Final Flashcards
Ratio of Perimeter
Scale Factor
Ratio of Area
(Scale Factor)^2
Ratio of Volume
(Scale Factor)^3
How to do Fraction on Calculator
- `Divide
- Hit Math
- Choose Frac
- Hit enter
Midpoint Formula
(x1 + x2) / 2 , (y1 + y2) / 2
Slope Formula
(y2 - y1) / (x2 - x1)
Distance Formula
d = √ (x2 - x1)^2 + (y2 - y1)^2
30 - 60 - 90 Shorter Leg
LL / √3 hyp/2
30 - 60 - 90 Longer Leg
sl x √3
30 - 60 - 90 Hypotenuse
sl x 2
45 - 45 - 90 Shorter Leg
divide by √2
45 - 45 - 90 Hypotenuse
sl x √2
A 45 - 45 - 90 is also an…
Isosceles Right Triangle
When multiplying radicals you…
- Multiply insides radical
- Multiply outside radical
- Simplify
Radicals only cancel if…
x√x not x√y
A tangent and radius are
Perpendicular
When adding number and another number with pi you…
Put the number without pi first in the equation
Perpendicular Slopes
Negative Reciprocoal
Parallel Slopes
Same slope
To divide radicals…
- If both inside radicals, just divide, and leave outside alone
- Or multiply by radical ex. √x / √x • x / √x = x√x /x = √x
What does midpoint do?
Split a line into two equal parts.
What does a bisector do?
Bisectors make two equal segments, only perpendicular bisectors make two equal, perpendicular halves.
The largest side faces the _____ angle
Largest angle
What is this triangle: a^2 + b^2 > c^2
The triangle is acute.
What is this triangle: a^2 + b^2 < c^2
The triangle is obtuse.
What is this triangle: a^2 + b^2 = c^2
The triangle is right.
What to do if sin (angle) = x / side
- (sin(angle) • x / side) side
2. (side) sin (angle) = x
HL Theorem
If the hypotenuse and the leg of one right triangle are congruent to the hypotenuse and the leg of another right triangle, then the triangles are congruent.
What to do if sin(angle) = side/x angle = 20 and side = 4
- sin(20º) = 4/x
- x(sin (20º) = 4/x)
- (sin20ºx = 4) / sin20º
- x = 4/sin20º
Supplementary means
It adds up to 180º
Complementary means
It adds up to 90º
Arc Length
nº / 360 • 2πr
Sector Area
πr^2 • central angle or n / 360
Circle on a plane
(x - h)^2 + (y - k)^2 = r^2
Center of Circle in Equation
(h,k) change the sign
Radius of Circle in Equation
r^2
When writing the equation of a circle for final answer remember:
- Graph it pre-squared
2. Square it in final answer
Complete Square and Get a Circle in Standard Form
- Gather x terms so they are next to each other in the equation. Get the constant (number without variable) on the other side of the equal sign.
- Look at the coefficients, number next to the variable, of the x and y term. Divide the coefficient to x by 2 and square it. Then add the number onto the inside of the equation, and on the other side of the equal sign. Divide the coefficient to y by 2 and square it. Then add the number onto the inside of the equation, and on the other side of the equal sign.
- Factor the x trinomial, diamond problem, then simplify it and plug the numbers into the diamond problem. Factor the y trinomial, diamond problem, then simplify it and plug the numbers into the diamond problem.
Standard Form of a Line
y = mx + b
If a line goes up left it is
negative
If a line goes up right it is
positive
Tangent Line
A line that intersects a circle in exactly one point.
The two segments tangent to a circle from a point outside the circle are…
Congruent
A segments whose endpoints are on a circle is called a
Chord
Diameter
A chord that goes through the center of a circle.
Minor Arc
Less than 180º
Major Arc
More than 180º
Semicircle
180º
Within a circle or in congruent circle,
- Congruent central angles have congruent chords.
- Congruent chords have congruent arcs.
- Congruent arcs have congruent central angles.
- Chords equidistant from the center are congruent.
- Congruent chords are equidistant from the center.
In a circle, a diameter that is perpendicular to a chord…
bisects the chord and its arcs.
In a circle, a diameter that bisects a chord (that is not a diameter)…
is perpendicular to the chord.
In a circle, the perpendicular bisector of a chord contains the…
center of the circle.
An angle inscribed in a semicircle is a…
right angle
The opposite angles of a quadrilateral inscribed in a circle…
add up to 180.
The measure of an angle formed by a tangent and a chord is…
half the measure of an intercepted arc. (the angle created by the line inside the circle and the line outside the circle is half of the arc that is in between the lines)
The measure of an angle formed by two lines that…
- intersect inside the circle is half the sum of the measures of the intercepted arc (but not the center)
- Intersect outside the circle is half the difference of the measures of the intercepted arcs.
For circle with triangle sticking out
- If you have one of two sides, add up the two numbers on the side with both numbers and multiply it by the outside number and square the number on the other side.
- If given both sides, side a and b. Add both sides and multiply by the outside and have them equal each other.
For a circle with and X on the inside…
Multiply both numbers that are on THE SAME LINE and put equal sign, and multiply both numbers on THE OTHER LINE.
Properties of a Parallelogram
- Opposite sides are congruent.
- Opposite angles are congruent.
- The diagonals bisect each other.
- Consecutive angles are supplementary.
(Must have at least one pair of opposite sides that are parallel and congruent to prove it to be a parallelogram)
Rhombus
- A parallelogram with two congruent, consecutive sides.
- Each diagonal of a rhombus bisects two angles of a rhombus.
- The diagonals are perpendicular.
A rectangle
- A parallelogram with at least one right angle
- The diagonals are congruent
Definition: parallelogram with one right angle.
A Square
- A rectangle and rhombus
- All properties of rectangle, rhombus, and parallelogram
- Diagonals of a square are congruent, bisect each other, each bisect 2 angles, perpendicular
Definition: a parallelogram, rhombus, and rectangle
It is a rhombus if…
- One diagonal of a parallelogram bisects two angles of the parallelogram
OR - The diagonals of a parallelogram are perpendicular
Definition: parallelogram with2 congruent, consecutive sides.
It is a rectangle if…
If the diagonals of a parallelogram are perpendicular
Trapezoid Midsegment Thm
The midsegment of a trapezoid is parallel to the base and the length of the midsegment is half the sum of the lengths of the bases.
Trapezoid
- parallel sides are bases
- non parallel sides are called legs
- two angles that share a base are called base angles
- isosceles trapezoids - legs are congruent
- two angles that share a leg are supplementary
- the base angles of an isosceles triangle are congruent
- the diagonals of an isosceles trapezoid are congruent
Kite
- two pairs of consecutive/adjacent congruent sides
- the diagonals are perpendicular
Isosceles Right Triangles
- two equal length legs
- 45 - 45 - 90
Equilateral Triangle
- Three equal sides
- All three angles are 60º
- It is a regular polygon with three sides.
- A perpendicular bisector splits it equally
Triangles and lines add up to
180º
Quadrilaterals add up to
360º
Isometery
Preserves size/distance of shape, image and original shape are congruent
Translation
- Isometry
- Moves figure, left, right, up, or down
Reflection
- Isometery
- A flip, reflected over line of reflection.
Reflection across line y = x
Switch x and y coordinates
Reflection y = -x
Switch numbers and change signs
Reflection over another line
move your axis to that point
Rotation
Turns figure around center of rotation.
Angle of Rotation
The number of degrees a figure rotates.
90º Rotation
(x,y) become (-y, x)
180º Rotation
(x,y) becomes (-x, -y)
270º Rotation
(x,y) becomes (y, -x)
90ª counter clockwise =
270º clockwise
270º counter clockwise =
90ºclockwise
Rotation when center is not (0,0)
- Make new axis around point
- Rename points as if your new graph was a normal graph
- Rotate
- Rename your points back to what they would be on a normal graph
Dilation
- Not an isometry
- Transformation when figure gets bigger or smaller
- The og figure is the scale factor
- makes similar triangles
- if scale factor > it is an enlargement
- if scale factor < it is a reduction
How far point from center for dilation?
- Multiply Scale Factor
2. Plot Point
Find the center of dilation:
- Draw lines connecting each point to its image
2. Find meeting point
Angle of Depression
- angle on top, outside
- the angle formed by the line of sight and the horizontal plane for an object below the horizontal
Angle of Elevation
- bottom angle, inside
- angle between the horizontal line of sight and the line of sight up to an object
Median: The medians of a triangle are…
concurrent at a point that is 2/3 the distance from each vertex to the midpoint of the opposite side.
Median: The point of concurrency is called the…
centroid
Median
is a segment of a triangle whose endpoints are a vertex and the midpoint of the opposite side.
