Mock Flashcards
What is the equation of a line?
Y=mx+c
M= Gradient
C= Cuts the y axis
How do you get the gradient of a line?
- Draw across the line then draw up to create a triangle
- Vertical distance/ horizontal distance=gradient 2/1=2
You can leave it in fraction form (gradients can be negative)
Important: if you get a gradient of 1 or -1 write it as x and -x.
What do you do if you get an question like (line equations):
Use the graph to obtain a formula for c in terms of n?
This means it wants you find the equation of the line, y axis is c and x axis is n
Change the line equation formula
y=mx+c
To
C=mn+c (then continue as normal)
Fractions to decimals conversions?
1/2= 0.5
1/4=0.25
3/4=0.75
1/3=0.33… (recurring)
2/3= 0.66… (recurring)
1/10= 0.1
2/10= 0.2
5/10= 0.5
etc..
1/5= 0.2
2/5= 0.4
3/5= 0.6
4/5= 0.8
How to change fractions to decimals?
Numerator/ Denominator= decimal version
How to change decimals to fractions?
!Always check to simplify!
0.7= 7/10
0.24= 24/100 = 12/50= 6/25
0.125= 125/1000=25/200= 1/8
What does deposit and balance mean?
.Deposit= upfront money
.Balance= the rest of money needed
Percentage and what do you divide them buy?
1%= divide by 100
10%= divide by 10
Percentage multipliers (increase or decrease)? (Without calculator)
Percentage multipliers (increase or decrease)? (With calculator)
1 way
Percentage / 100% x value = extra value
Value- extra value= total value
2 way
100%+ other percentage = total percentage
Total percentage of value
Total percentage/100 % x value= total value
Congruent triangles?
SSS= side side side (same)
SAS= side angle side (same)
ASA= Angle side Angle (same)
RHS= Right-angle hypotenuse side (same + two right angles + longest side)
How to tell if a triangle has a hypotenuse?
The hypotenuse of a right triangle is always the side opposite the right angle.
How answer an exam question on Congruent triangles?
- Highlight and re draw the triangles mentioned in the question.
- Bulletpoint answers that’s shows they congruent give reasons
- Use information in question and in diagram
- Concluding statement
What is Standard form and how to write in standard form?
Ax10^power
A= alway a number between 1 and 10
How to write in standard form:
1. Put a decimal between the number to make below 10
2. Count from the decimal point to end= power
Standard form with decimal is already in the number?
When working with numbers with a decimal already, you count backwards to it.
Writing numbers which are already in standard form out of it you do?
- Write A out
- Then count backwards or forwards (moving decimal point) (depending if it’s negative or positive) based on the power
- Then fill in the 0s
- If it’s negative add a 0 after the new decimal point
Exam question on standard form?
- Get rid of the brackets
- All multiplication so you can change the order to make it easier
- Indices law add the indices don’t work them out
Exam question standard form?
- Get rid of brackets
- Not all multiplication so can’t do it in any order
- Turn into fraction
- Law of indices
Working with indices, what does these mean?
Evaluate
Calculate
Workout
All means = find the answer
All five index laws?
The fifth index law?
What is the power of this number: 5?
Numbers without a seen power always have the power of 1
Exam question using index rule 5?
Exam question using index notation 5?
Fractional indices using the fourth rule?
- Carry out the rule
- The root of the number is what you multiple itself root to get itself
- The extra square you add on
- Work it out= answer
The forth index rule using a hence question?
Hence= use the other question to help
What is the 6th index rule?
Combining rule 4 and 5!
Using the 6th index rule, what are the answers to theses:
- The minus in fraction means it’s 1/ number +power
- Change to a square root with no minus on the power
- Figure out the square root
Extra steps: (if needed)
1. Figure out the root ether cube or squared
2. Then figure out the power
Harder exam question using the 6th index rule?
Harder exam question:
1. Find a link between 32 and 16 to do with powers both in power of 2
2. Multiply powers and change the fraction to a number with a minus on the power
3. Powers are 5x=-4
4. X= -4/5
Area of a circle?
Area of a circle= π r²
What is the diameter and circumference of a circle?
Circumference= whole circle (2πr)
Diameter= x 2 radius
How to find the cumulative frequency?
- Use the frequency from the first diagram, the first one stays the same e.g 2 then the second one you add 2 and the next frequency number 6= 8
- Drawing the graph: always plot the end points (0<t<10) and the cumulative frequency number.
- The curve is a s shape all the time.
- You can find on the graph the Median (half) , Lower quartile (half of median), upper quartile (median+LQ) and interquartile range (UQ-LQ)
Using the graph to find a particular numbers of data? E.g Estimate the number of students who spent more than 42 minutes on their homework.
- Find 42 on the minutes axis
- Then draw up until you reach the line and draw across to the other axis
- Then read the value on that axis e.g 28, total pupils 30, 30-28=2
Answer= 2
Multiplying fractions?
multiplying numerators (top numbers) and denominators (bottom numbers) then simplify
. you can cancel before you multiply you can cancel any top number with any bottom number can be diagonal
Mixed numbers when Multiplying fractions?
- Turn the mixed number into a top heavy fraction (bottom number stays the same, then mixed numbers bottom number + top number= new top number)
- Check for cancellations
- Then multiply
- Change back into a mixed number (bottom number/ top= whole number, bottom stays the same, top number is what is left over from dividing to get the mixed number)
Multiplying decimals?
- Ignore decimal points the multiply the numbers
- Count the number of decimal places in question in both numbers then you will have same in the answer
- Check by rounding the question
Recurring decimals?
.Can be written with a dot on the numbers recurring, and if there is 3+ recurring numbers put a dot on first recurring number and on the end one to say all the ones in between also are recurring
Writing a recurring decimal as an exact fraction
.put a digit over the correct number of 9s, after you can simplify
How to prove a recurring decimal?
Proving recurring decimals
1. Say let x=the decimal
2. If it’s 1 recurring number you times by 10 if it’s 2 recurring numbers you times 100 ect…
3. You then subtract the 1st line from the second line
4. Turn into a fraction, simpfly if you can
Harder exam question recurring decimals?
.If you do all the steps and get a decimal you must multiply it out
What are prisms?
a solid geometric figure whose two ends are similar, equal, and parallel rectilinear figures, and whose sides are parallelograms.
(Not circular)
What is the volume of a prism?
A prism has the same cross section all the way through.
E.g cuboid, triangular prism
- Shade in the cross section
- Find the are of the cross section e.g triangle A= bxh/2
- Multiply the area by the length
Volume of a prism= cross section x length
Cuboids and cubes surface area?
- Area of each individual surface (draw dotted lines onto the shape)
- The sides are Rectangles: LxW
- Layout: Bottom, top, right, left, front, back
- Units: cm^2
Cuboids and cubes volume?
Volume of a cuboid:
Volume: length x width x height
Volume of a cube:
Volume= area^3
What is Direct proportional?
.Direct= as one increase the other also increase
What is inverse proportion?
.Inverse= as one increases the other decreases
What does it mean in exam if they say the word proportional?
If it says proportional= direct proportional
How to answer direct proportional questions?
- y ∝ x
- y=kx k= (constant)
- Swap the letters for the numbers given
- To figure out k (constant)
- Then you write an equation connecting y and x using y=kx then swap k for the figured out constant
Extra steps when it’s asks you find another value of a letter e.g find value y when x=8 - Substitute the the new value, then work out the value
How to answer a inversely proportional question?
- y ∝ 1/x
- y= k/x
- Swap the letters for the numbers given
- To figure out k (constant)
- Then you write an equation connecting y and x using y=k/x then swap k for the figured out constant
Extra steps when it’s asks you find another value of a letter e.g find value y when x=4 - Substitute the the new value, then work out the value
exam question on direct or inverse proportion?
exam question on direct or inverse proportion?
How to use a tree diagram?
(can be decimals or fractions)
1. Draw the options
2. Probability must add up to 1 so spilt between options (on each branch)
3. Then repeat the options
4. Then you can work out the probability, by going along a branch and multiplying
5. For the total probability is to add the two probability’s together
Tree diagram exam question?
(You can work out what happens if she wins both then take away from 1 to get probability of not winning both)
Harder tree diagram question?
Probability?
Using fractions:
Denominator= total options
Top number= chance of happening to a particular option
‘And’ ‘or’ probability questions?
Example of an ‘and’ question?
Exam question ‘And’ and ‘or’ questions mixed?
Hard probability ‘and’ and ‘or’ question?
- Work out the probability that they get the same
- Then take it away from 1 (total probability) answer if they not the same chocolate
What can you work out from a speed-time graph?
What can you work out from a velocity time graph:
1. Acceleration at any given time
2. Total distance travelled at any given time
How to find the acceleration?
.measured in= m/s^2
.horizontal= 0 acceleration (constant)
.Find Gradient (acceleration) m= change in y/change in x
How to find the total distance traveled?
Separate graph into shapes then find the area of each shape the add it up
Or
If it’s parallel (top and bottom line) you can use the formula for the area of a trapezium
Harder question: (non constant acceleration)? (Curved graph)
Estimate for acceraltion at a certain time:
1. Find the point
2. Draw a tangent (where the point is touched by a line but the rest isn’t)
3. Draw a right angled triangle using the line to find gradient
4. Work out the scale on graph
5. Find the gradient change in y/ change in x
(Hard question) Finding the average speed? (curved graph)
- Speed= distance/time
- Distance travelled under the graph (cut up Into shapes) (Trapezium parrel lines are a and b)
- Add up the areas
- Time is last number on the time axis
- The divided them Speed= distance/time
Worded question: speed time graphs (hard)?
(straight lines are constant)
1. Use the words to create a graph
2. Work out the areas of each shape then add them together
What is a Venn diagram and how do you figure on out?
.The people who has nothing go on the outside of the circle (the take away the people from total).
.to figure out who has both you add up the ones that say they have the items then take away away from the actual total of people = the ones that have both.
.to find who goes in the individual ones you minus the total you are told from the ones that have both.
(To check you add them up to check they match the total)
What are the notations used in Venn diagrams? (3)
Exam question on notation in Venn diagrams?
Probability using Venn diagrams?
(In probability you don’t need to cancel down fractions.)
Harder question on Venn diagram propbablitys?
Solving equations with fractions (method 1)?
You can multiply out the fractions
Solving equations with fractions (method 2)?
Multiply both sides by the fractions recipcaol e.g 2/3 (3/2)
Fractions of a number e.g 3/2 8 (8/2=4x3) divide by the bottom number times by the top
Solving equations with fractions with two fractions?
Multiply both sides by the common (same) denominator (use the least) to each thing the equation
Trigonometric values?
Negative numbers: Adding and subtracting?
.Two signs next to each other are different you take away
.two signs are the same you add
Negative numbers: Multiplication?
.two signs are different answer will be negative
.two signs are the same the answer will be positive
Negative numbers: one sign in the middle?
.C and d you swap around the numbers then add a minus on to the answer
How to Factories quadratics?
- Two brackets
- X^2= two xs in each bracket
- The second number, needs the two numbers in the brackets to add up to it
- The third number, need the two numbers in the brackets to multiply up to it
You can check my expanding it by doing the moon shape: the first bracket each thing in it multiplied by the other bracket
How to factories Negative coefficients?
- Start in the same way 2 brackets that both have an x in it.
- If it is minus when multiplying you need one minus
- If both are you still need one minus just make sure they add and multiply to the correct numbers
When do you use two minuses?
Sometimes you can make both minuses if you need a negative adding number but a positive multiplication number, because multiplying two minuses makes a positive but add to a minus.
How to solve quadratics?
The rights hand equals 0.
If you have two things that multiply to equal 0, then one of them is 0.
1. Put the equation out brackets (separate)
2. Then solve the linear equations, the answers will be the solutions
You can check by substituting each answers to the solutions.
How do you Solve when it’s not 0?
Make it equal zero by subtracting or adding it to both sides.
How Factorise difficult quadratics?
The the coefficient of x^2 is larger then
1. Take the coefficient of x^2 and multiply it by the end number in the equation (constant)
2. Then you have to find two numbers that multiply to make the answer to the first step and also need to add to make the middle number.
3. You the spilt the middle number into the answer to the second step also including x to each of the answers
4. Then put the first number and the last number next to the spilt number.
5. You then factorises (normally) the two terms each (two total)
6. You get the same bracket from the two factorisation, this is the first bracket of the factorisation of the difficult quadratic
7. The second bracket of the factorisation of the difficult quadratic, is from the left overs.
Larger number multiplication Factorising difficult quadratics?
Larger number multiplication: e.g 168 you half the number as you go up faster (1 and 168, 2 and 84)…
How to Solving quadratics with coefficient of x^2 ?
- Factorising as normal with quadratics coefficient of x^2, so you have brackets
- Then get rid of the brackets and just have the two linear equation that equal 0
- Then solve the two linear equations to find the two values of x (can be in fractions)
All the 4 different algebraic equations?
Formulae exam question?
Equations and expressions exam question?
You can check by putting the value of x into the diagram.
What are identities and how to show they are true?
You can’t move terms from one side of the equation to the other
To show it’s true: Make the left hand side look exactly like the right hand side
1. Simplify the left side, getting rid of brackets(expanding) and collecting alike terms
2. And do any thing else needed to make it the same e.g factorising
Finding the value of a letter in identity’s?
- Expand and simplify in any way
- Look at all the x terms on both sides and form an equation
- Then solve the equation for the values of a
- Then use the number terms and the d terms and form an equation
- Then solve the equation for the value of d
You can check by putting the values of a and d into the smaller equations to see if it equals the same.
Fibonacci sequences?
You add the the two past numbers together to get the next two terms e.g 1+1=2 1+2=3
Fibonacci sequences Harder questions ?
X1= means the first number in the pattern
X2= means the second number in the pattern
Xn>300 = what is it’s number (n) in sequence when the value first goes over 300.
Fibonacci sequences Harder questions ?
Xn+Xn+1=21= means the 21 (value) in sequence
Fibonacci sequences Harder questions ? (Calculator)
Special sequences: what is a term to term rule?
A term to term rule= means how do get from one number to the next
Square numbers- special sequences?
Times a number by the same number e.g 2x2=4 is a square number
The nth terms is n (which is a placement in the sequence) then what you do to get the next number.
Triangular numbers- special sequences?
Finding the nth term of triangular numbers?
You can check by implementing this into one of the patterns.
What are the conditions for Quadratic graphs?
Drawing a quadratic graph?
X^2= a curved graph not a straight line
Completing the table for the equation
Non calculator-
1. Substitute the letter x e,g 0
2. 2x0^2-4x0-2 = 0-0-2= -2
3. Now 2: 2x2^2-4x2-2 (BIDMAS) 8-8-2=-2
4. Check for patterns then complete the rest
After completing the table you can plot the points on to the graph, and join the points with a smooth curve. Important: The bottom of the graph must be curved.
Finding values from a quadratic graph?
E.g use your graph to find the value of y when x=1.5
1. Draw on the graph a line straight through 1.5 x
2. Where the line crosses the graph and read the value on the y axis with another drawn line
3. Answer: y=-3.5
Important: You can get multiple values on these questions
Exam question- Question C?
- To solve this because it is different to the graph because of the -3 you must prove it true
- y= x^2-2x
- X^2-2x-3=1
- To make it equal and the same as the graph you need to move the -3 so it is on the equals side
- X^2-2x=4
- y=4 draw this line on the graph
- Then draw down from where it crosses the line to the get the x values
- X=1.25, 3.25
Examples of quadratics?
Positive: U
Negative: n
What are all the parts of a quadratic graph?
Y intercept is the consent e.g -3 can also be called the value of something
Inequality regions and the method?
Y= line across
X= line down
<>= bigger than signs mean to draw a broken line
<> (with a line underneath) =equals a unbroken lines
Method:
1. Shade in the area that’s is not going to be discard(when doing multiple)
2. When having an inequality like y<x+1 you draw a a x y table
3. Then to check which area satisfies the inequality take a coordinate on one of the sides and put it into the inequality e.g y<x+1 put coordinates (1,4) 4<1+1 = 4<2 (which is not true) so you shade the region you got that coordinate from
4. If you have an inequality like x+y<5 you need to make y the subject to so you can do the y and x table to plot the line, you then pick a coordinate from a region and test it in an inequality in the unchanged/ original one.
Graph transformations all 4?
Y=f(x) is the normal graph without change
Exam question
Exam question
Cubic graphs?
Cubic y=x^3
Graphs are a wiggle
To be cubic the highest power of x needs to be cubed.
Turing point= where the graph turn and changes direction
If it -x^3 it switches side
Exam style question:
- Complete the table as usual (start with 0 it’s easier)
- The turning point is the curve (change in direction) only put the x axis point down not y axis
reciprocal graphs?
reciprocal y= 1/x
They are asymptote to the curve= this means the curve will get closer and closer to them but never touch them
Values of a reciprocal graph?
Exam style question of a reciprocal graph?
Addition and subtracting vectors?
.Length of vector is the magnitude and the direction is where it is pointing
.If you have two vectors a and b draw another line next to them to create a triangular shape saying a+b, if it’s subtraction it goes in the other direction
Examples of addition and a combination of vectors?
.vector 2(a+b)= 2a+2b draw the two a lines separate and then the two b lines separate then a line to join them like a triangle saying 2a+2b
2(a+b) is a scalar multiple of a +b: meaning it has the same direction
Multiply vectors?
Vector notation?
Example of vector notation?
.express a line e.g x and y you put a arrow over head of xy going in the same direction
.underline your vectors (a vector is the letter given for one the notions)
.if you want to change it so it’s goes in opposite direction yx then the vector will change to be negative
.combing vectors going the long way round go from xz to zy which means your at xy, to show this you minus the vector of xz and zy then put = to the vector of xy
Vectors and ratio?
.find a path to get to the vector
.the ratio add them to get denominator of the other part of of the vector line e.g BD= 1/4 BC
.get to BC by going the opposite way to reach the edges of BC to get the expression
.subsuite expression of BC
.expand any brackets
.the original path way at the start to find AD subsuite the values for AB and BD then work it out to get AD
Column Vectors?
Vectors are often split up into two parts, which we call components: An x component, which moves left or right, and a y component, which moves up or down.
Adding and subtracting column vectors?
To add/subtract column vectors, we add/subtract the
x and y values separately.
Multiplying Column Vectors?
Properties on solids: cuboid, cube, triangular prism, square-based prism, triangle- based pyramid: faces, edges, vertices?
.Prisms: have the same cross-section all the way through
E.g cylinder, triangular prism, cuboid, cube
Cross section: the shape it looks like at the front e.g circle, triangle
Name 8 prisms?
Introduction to Proportion odd numbers?
.if one side gives a really bad decimal change your working out to the other side to get an easier number to work with
.remember if a time and the number of people increase the time will decrease so one side you multiply the other you divide
Choosing correct statements:ratios and fractions?
.check through (do not add them for the denominator for this question) the last number is the denominator the first is numerator
Ratio and graphs?
.Ratio and graphs, plot the ratio as if it’s x,y
Conversion of measurements: length, mass, capacity?
clockwise and anti-clockwise direction?
dividing fractions?
KFC
The 6 types of triangles and their properties?
The 3 Types of angles?
Square numbers from 1-30?
Square roots of: 1, 4, 9,16,25, 36, 49, 64,81, 100, 121, 144, 169, 196, 225, 256, 289, 324,361, 400, 441, 484, 529, 576, 625?
Prime number 1-500?
Cubes 1 to 10?
How to change the subject of a formula?
.do the opposite of the to move it to the other side
Opposites:
Cubbed to cubbed root
Squared to square root
Add to subtract
Multiply to divide
Hard questions: changing subject formula?
.Move the easy terms first
.if you multiply an side by a term you must put the other terms in brackets
Changing subject of the formula with brackets?
.if the term you want is in brackets you must multiply it out
Changing subject of the formula with fractions?
.if it’s in a fraction multiply out the fraction
What happens when the term you want to make the subject appears multiple times?
.get all the ones with the desired terms on one side and all the others on the other side
(Put the desired one with side with more on)
. terms with multiple desired terms without numbers you can factorise it
Algebraic fractions- Simplifying?
Algebraic fractions Straight forward cancelling?
.subtract powers and divide numbers
Algebraic fractions Factorise then cancel?
.can be quadratics so factorise, you can tell if it’s a quadratic by the x^2
.or if it’s not a quadratic you can factorise normally
.if it’s difference of two squares you can factorise again: (must have a minus in between)
Algebraic fractions- Adding or subtracting fractions
.the denominator must be the same, and do what you do to the bottom to the top
- Must have a common denominator (what you do to bottom you do to the top)
- Join the fractions an if minus symbol you must put it on the second top fraction (if there is already a minus on a number it’s turns to a plus
- Multiply out the brackets then join add/subtract like terms (simplify)
- Then if you can’t do anything else that should be the answer
Algebraic fractions harder questions- addition?
.the denominators are letters so times each denominator by the other letter then the same to the top (you have to use brackets if there is more then one term when multiplying)
.don’t multiply out brackets if it won’t simplify even more
Harder questions algebraic fractions- quadratic in the denominator?
.Factorise it if you can (difference of two squares), then you can make both denominators the quadratic
.whether you have a quadratic you must factorise it
Solving: algebraic fractions?
- Write the other side as a singular fraction
- The solve how you normally solve to get the value of x e.g multiplying, adding, subtracting
- Solving linear equations
Algebraic fractions - hard question solving quadratics in fractions when it is equal to 0?
Solving, quadratics when it = 0
1. Make the one side one hole fraction
2. Then multiply the bottom the fraction the singular number on the other side
3. Expand the brackets (quadratic)
4. Then move all terms to the left so the other side is worth 0
5. You then have a quadratic solving question
6. Factorise the quadratic
7. Then separate each bracket saying it =0 to make 2 linear equations
8. Then continue to solve as normal to get what x equals (two answers)
Rule one of Surds?
.multiply the same root number will equal taht number without the root
Rule two of Surds?
.Pick numbers that can simplify (pick the biggest square that goes into the number)
.if you undo the root e.g 9 Square root to 3 then do it
Rule 3 of Surds?
.can find a square number or divide the two roots
Exam style question: Surds?
Adding and subtracting Surds?
.common rules on Algebra apply to Surds
Simplifying surd expressions? (Rule two)
(Show clearly that:) questions?
.ignore the answer beside it and continue normally
Exam style questions?
Exam style question?
Expanding brackets surd expressions?
.do it in the same way you expand a quadratic normally and remember to use the rules and simplify
Integers= whole number
Rationalising the denominator Surds?
This means you need to get the square root out of the denominator
1. Multiply the root by itself in a fraction both as the numerator and denominator
2. On the top you should have a number and the root, then the same root on the bottom twice
3. Rule 1 means the two same roots equal the number in it so it changes the denominator to a number
4. You then divide the denominator by the singular number on the numerator
5. The answer will be the root and the divided number
Hard exam questions surds?
.first you must multiply the root then add the number in its roots
Difficult Rationalising Surds-The difference of two squares?
The difference of two squares
Difficult Rationalising?
- Multiply the whole fraction by the denominator both top and bottom
- Then lay out the new fraction
- Complete any subtracting, adding, multiplying diving in the fraction
- Then lay it out after
How to show that something can written as something else: Surds (rationalisation)?
- Rationalise the denominator
- Combine into a normal fraction
- Complete any subtracting, adding, multiplying diving in the fraction and expand the brackets
- Rewrite the roots if necessary with the rules
- Complete any subtracting, adding, multiplying diving in the fraction
Loci and constructions- Parallel?
Parallel lines never meet.
Loci and constructions- Perpendicular lines?
Perpendicular lines are at right angles. There is a 90° angle between them.
Loci and constructions- Vertex?
A corner or a point where two lines meet.
Loci and constructions- Angle Bisector?
Angle Bisector: Cuts the angle in half.
1. Place the sharp end of a pair of compasses on the vertex.
2. Draw an arc, marking a point on each line.
3. Without changing the compass put the compass on each point and mark a centre point where two arcs cross over.
4. Use a ruler to draw a line through the vertex and centre point.
Loci and constructions- Perpendicular Bisector?
Perpendicular Bisector: Cuts a line in half and at right angles.
1. Put the sharp point of a pair of compasses on A.
2. Open the compass over half way on the line.
3. Draw an arc above and below the line. 4. Without changing the compass, repeat from point B.
5. Draw a straight line through the two intersecting arcs.
Loci and constructions-Perpendicular from an External Point?
The perpendicular distance from a point to a line is the shortest distance to that line.
1. Put the sharp point of a pair of compasses on the point.
2. Draw an arc that crosses the line twice. 3. Place the sharp point of the compass on one of these points, open over half way and draw an arc above and below the line.
4. Repeat from the other point on the line.
5. Draw a straight line through the two intersecting arcs.
Loci and constructions- Perpendicular from a Point on a Line?
Perpendicular from a Point on a Line
Given line PQ and point R on the line:
1. Put the sharp point of a pair of compasses on point R.
2. Draw two arcs either side of the point of equal width (giving points S and T)
3. Place the compass on point S, open over halfway and draw an arc above the line.
4. Repeat from the other arc on the line (point T).
5. Draw a straight line from the intersecting arcs to the original point on the line.
Loci and constructions- Constructing Triangles (Side, Side, Side)?
- Draw the base of the triangle using a ruler.
- Open a pair of compasses to the width of one side of the triangle.
- Place the point on one end of the line and draw an arc.
- Repeat for the other side of the triangle at the other end of the line.
- Using a ruler, draw lines connecting the ends of the base of the triangle to the point where the arcs intersect.
Loci and constructions- Constructing Triangles (Side, Angle, Side)?
- Draw the base of the triangle using a ruler.
- Measure the angle required using a protractor and mark this angle.
- Remove the protractor and draw a line of the exact length required in line with the angle mark drawn.
- Connect the end of this line to the other end of the base of the triangle
Loci and constructions- Constructing Triangles (Angle, Side, Angle)?
- Draw the base of the triangle using a ruler.
- Measure one of the angles required using a protractor and mark this angle.
- Draw a straight line through this point from the same point on the base of the triangle.
- Repeat this for the other angle on the other end of the base of the triangle.
Loci and constructions- Constructing an Equilateral Triangle (also makes a 60° angle)?
- Draw the base of the triangle using a ruler.
- Open the pair of compasses to the exact length of the side of the triangle.
- Place the sharp point on one end of the line and draw an arc.
- Repeat this from the other end of the line.
- Using a ruler, draw lines connecting the ends of the base of the triangle to the point where the arcs intersect.
Loci and constructions- Loci and Regions?
A locus is a path of points that follow a rule.
For the locus of points closer to B than A, create a perpendicular bisector between A and B and shade the side closer to B.
For the locus of points equidistant from A, use a compass to draw a circle, centre A.
For the locus of points equidistant to line X and line Y, create an angle bisector.
For the locus of points a set distance from a line, create two semi-circles at either end joined by two parallel lines.
Loci and constructions- Equidistant?
A point is equidistant from a set of objects if the distances between that point and each of the objects is the same.
Ratio- advanced (standard questions)?
Standard questions
1. Write it out in the normal ratio way (set out headings)
2. Multiply up the numbers up so they all on the same ratio
3. Add them up
4. The total divided by the total ratio number
5. The answer to step 4 multipled by the ratio for that thing
(.when a question says e.g ‘five times as many blue discs as green discs’ the ratio is blue:5 green:1)
ratios-advanced (standard question lines)?
.You need the same amount of parts so you must multiply to get it
Equations to ratios?
.if you multiply the outers together you get what it was previously and if you multiply the inerds you get that also
.sometimes you’ll need to collect like terms
Hard questions- ratios to equations?
Ratios to equations?
.if you have two ratios that are equal to each other you can form an equation, you can do this by multiplying the outers to outers and inners to the inners
.if the questions say ‘show that’ whatever is on the equal side you want to isolate and sometimes further on you have to factorise your make it easier to divide and single out the term you want
.if it’s minuses and you want it to be positive you can multiply by 1
Ratio to equations- quadratics?
Ratios- another type of question?
Ratios can be placed in fractions: a/b = 1/4
