8. Geometry Flashcards

Question

What is a Cube?

Answer 1

A three-dimensional shape consisting of six identical faces, all of which are squares

Answer 2

Surface area = the area of any one face multiplied by 6. NOTE: only need to know length of one side to determine surface area

Answer 3

Volume = s^3, where s refers to the length of any one side of the cube NOTE: only need to know length of one side to determine volume

Answer 4

BEWARE OF THE GMAT VOLUME TRICK. It is tempting to answer 50 books. However, this is incorrect, because you do not know the exact dimension of each book! One book might be 5*5*4 while another book might be 20*5*1 – both of which have a volume of 100 in^3. However, both have different rectangular shapes Takeaway: -When you are fitting 3-dimensional objects into other 3-dimensional objects, knowing the respective volumes is not enough – you must know the specific dimensions (length, width, height) of each object to determine whether the objects can fit without leaving gaps

Answer 5

``` Area(EFGH) = Area(ABCD) – Area(EFGH) 2Area(EFGH) = Area(ABCD) 2Area(EFGH) = 6*6 Area(EFGH) = 18 = x^2 X = √18 = 3√2 ```

Answer 6

- A three-sided closed shape composed of straight lines | - The interior angles add up to 180°

Answer 7

Area of triangle = (1/2)(b)(h) - Base refers to the bottom side of the triangle (note that ANY side of the triangle could act as a base) - Height always refers to a line drawn from the opposite vertex to the base - The base and the height must be perpendicular (form 90 degree angle) to each other *The height can be outside the triangle! (You just have to extend the base, but not for the purposes of calculating the base in the area formula)

Answer 8

An "angle" or place where two lines of a shape meet; for example, a triangle has three vertices and a rectangle has four vertices

Answer 9

The smaller sides of a triangle; usually used in describing a right triangle, in which there is one hypotenuse (the longest side) and two legs (the shorter sides)

Answer 10

The longest side of a right triangle. The hypotenuse is opposite the right angle.

Answer 11

- A triangle that includes a 90°, or right, angle - Given the lengths of any two of the sides of a right triangle, you can determine the length of the third side using the Pythagorean Theorem - With 30-60-90 and 45-45-90 right triangles, you only need the length of one side to determine the lengths of the other sides *Right triangles are essential for solving problems involving other polygons

Answer 12

(1) Sum of the three angles of a triangle equals 180 (2) Angles correspond to their opposite sides - Largest angle is opposite the longest side, while the smallest angle is opposite the shortest side and vice versa - Additionally, if two sides are equal, their opposite angles are also equal (isosceles triangles)

Answer 13

If you are given two sides of a triangle, the length of the third side must lie between the difference and the sum of the two given sides - Any side of a triangle must be less than the sum of the other two sides - Any side of a triangle must be greater than the difference of the other two sides

Answer 14

The sum of the internal angles of a triangle must add up to 180 degrees. As a result, if you know two angles of a triangle, you can find the third angle Or you might be given one angle and the other angles in terms of x, in which case you solve the equation for x

Answer 15

- Any triangle in which one of the angles is a right angle (90 degrees) - Every right triangle is composed of two legs and one hypotenuse (side opposite the right angle, or c) * APPLY PYTHAGOREAN’S THEOREM TO DETERMINE LENGTH OF SIDES

Answer 16

a^2 + b^2 = c^2 -The lengths of the three sides of a right triangle are related by the equation above, where a and b are the lengths of the sides touching the right angle, also known as legs, and c is the length of the side opposite the right angle, also known as the hypotenuse * Only applies to right triangles * You can always find the length of the third side of a right triangle if you know the lengths of the other two sides

Answer 17

``` A subset of right triangles in which all three sides have lengths that are integer values a-b-c 3-4-5 or 6-8-10 or 9-12-15 or 12-16-20 5-12-13 or 10-24-26 7-24-25 8-15-17 9-40-41 ``` Note: You can double, triple or otherwise apply a common multiplier to these lengths

Answer 18

A non-right triangle with two sides equal to two parts of a Pythagorean triplet Example: a triangle with one side equal to 3 and one side equal to 4 does NOT necessarily mean that the third side has a length of 5; this rule only applies to triangles that are KNOWN to be right triangles

Answer 19

- A triangle that has two equal angles and two equal sides (opposite the equal angles) - An isosceles right triangle has one 90 degree angle (opposite to the hypotenuse) and two 45 degree angles (45-45-90 triangle)

Answer 20

If the angles of a triangle are equal to 45, 45 and 90 degrees, then the lengths of the sides are proportional to x : x : x√2 Leg : Leg : Hypotenuse X : X : X√2 *IMPORTANT: an isosceles triangle is equal to exactly one half of a square (i.e. two right isosceles triangles make up a square)

Answer 21

A triangle that has 3 angles (all 60 degrees) and 3 equal sides

Answer 22

- If one length is equal to the diameter of the circle, then the triangle is a right triangle - The angle opposite to the hypotenuse is equal to 90 degrees - Otherwise the triangle is not a right triangle

Answer 23

If the angles of a triangle are equal to 30, 60 and 90 degrees, then the lengths of the sides are proportional to x : x√3 : 2x Short Leg : Long Leg : Hypotenuse X : X√3 : 2X

Answer 24

- The side of an equilateral triangle is the same as the hypotenuse of a 30-60-90 triangle - The height of an equilateral triangle is the same as the long leg of a 30-60-90 triangle - Thus, the long leg has a length of 5√3, which is the height of the equilateral triangle

Answer 25

In addition to the standard area formula for triangles, equilateral triangles have a special formula for area: Area = (√3 * S^2)/4, where S is the length of any side of the equilateral triangle.

Answer 26

Set the opposite angles equal to each other

Answer 27

Set the opposite sides equal

Answer 28

d = s√2, where s is a side of the square *This can also be the face diagonal of a cube

Answer 29

- The main diagonal of a cube is the one that cuts through the center of the cube - The diagonal of a face of a cube is not the main diagonal d = s√3, where s is an edge of the square

Answer 30

``` √60 = s√3 s = √60 / √3 = √20 ```

Answer 31

To find the diagonal of a rectangle, you must know either the length and width or one dimension and the proportion of one to the other Using Pythagorean’s theorem, c = √(a^2 + b^2)

Answer 32

- The main diagonal of a rectangular solid is the one that cuts through the center of the solid - The diagonal of a face of the rectangular solid is not the main diagonal - The main diagonal of a rectangular solid can be found by using the “Deluxe” Pythagorean Theorem: d^2 = x^2 + y^2 + z^2, where x, y, and z are the length, width, and height of the rectangular solid, and d is the main diagonal

Answer 33

-Triangles in which the three corresponding angles are identical and the corresponding sides are in proportion

Answer 34

(1) 2 Angles: It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180 degrees (2) If two similar triangles have corresponding side lengths (or heights or perimeters) in ratio a:b, then their areas will be in ratio a^2:b^2

Answer 35

For similar solids (e.g. quadrilaterals, pentagons, etc) with corresponding sides in ratio a:b, their volumes will be in ratio a^3:b^3

Answer 36

(1) Can find the third side, either by using the full Pythagorean theorem or by recognizing a triplet (2) Make that unknown side less than the sum of the other two sides but more than the difference

Answer 37

1 to 4. You can use the shortcut or calculate the Area for each triangle in terms of x. Ratio of Areas = a^2 + b^2 = 1^2 : 2^2 = 1 : 4 Area(A) = (1/2)(x)(2x) = x^2 Area(B) = (1/2)(2x)(4x) = 4x^2 Ratio of Areas = 1:4

Answer 38

If AD = DB = DC, then the three triangular regions in this figure are all isosceles triangles. Therefore, you can fill in some of the missing angle measurements. Since you know that there are 180 degrees in the large triangle ABC, you can write the following equation: x + x + 20 + 20 + 60 + 60 = 180 2x + 160 = 180 x = 10

Answer 39

Area Equilateral Triangle = (√3 * S^2)/4 = 16√3

Answer 40

Look at the star as one big triangle and three smaller triangles. Area Large Triangle = (1/2)(12)(6√3) = 36√3 Area Small Triangle = (1/2)(4)(2√3) = 4√3 Area Star = 36√3 + 3(4√3) = 48√3

Answer 41

- Set of points that are equidistant from a central point - A circle contains 360 degrees ``` Circumference = 2(pi)(r) Area = (pi)(r^2) ```

Answer 42

Half of a circle; a semi-circle contains 180º, exactly half of the 360º in a circle.

Answer 43

- Any line segment connecting the center and any point on the circle is a radius (usually labeled r) - All radii in the same circle have the same length

Answer 44

-Line segment that passes through the center of a circle and connects two opposite points on the circle (usually labeled d) d = 2 * r

Answer 45

- Any line segment that connects two points on a circle | - Any chord that passes through the center of the circle is called the diameter

Answer 46

(1) C = circumference = 2(pi)(r); distance around a circle (2) d = diameter = 2*r; distance across a circle (3) r = radius = d/2; half the distance across a circle (4) A = Area = (pi)(r)^2; area of a circle **If you know any of these values, you can determine the rest

Answer 47

Measure of the distance around a circle C = 2(pi)(r) OR C = (d)(pi)

Answer 48

The space inside the circle (usually labeled A) A = (pi)(r^2)

Answer 49

Volume = (4/3)(pi)(r^3)

Answer 50

Revolution = Circumference A point on the edge of a wheel travels one circumference in one revolution

Answer 51

If you know one thing about a circle (e.g. area), then you can find out everything else about the circle by using the standard formulas (r, d, and C)

Answer 52

Any fractional portion of a circle (i.e. a slice of the pie)

Answer 53

The proportion of the central angle to 360 is the same as the proportion of the area of the sector to the area of the circle For example, if the central angle is 60, the proportion is 60/360 = 1/6. The sector, then, is 1/6 the area of the total area

Answer 54

The remaining portion of the circumference included in the sector Arc Length = Central Angle/360 * 2(pi)(r)

Answer 55

- An angle whose vertex lies at the center point of a circle - The degree measure between the two radii that form a sector - The central angle tells you the fractional amount that the sector represents; but any of the three properties of a sector (e.g. central angle, arch length, and area) could be used Central Angle / 360 = Sector Area / Circle Area = Arc Length / Circumference

Answer 56

Perimeter of sector = 2(r) + Arc Length

Answer 57

- An angle that has its vertex on the circumference of the circle itself, and thus does not originate from the center of the circle - The other two endpoints define the intercepted arc on the circle

Answer 58

The inscribed angle is exactly half of the central angle

Answer 59

The two inscribed angles are congruent

Answer 60

- A triangle is said to be inscribed in a circle if all of the vertices of the triangle are points on the circle * RULE: If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle; conversely, any right triangle inscribed in a circle must have the diameter of the circle as one of its sides (thereby splitting the circle in half)

Answer 61

Right triangle

Answer 62

Figure out the fraction of the circle that the sector represents ``` Fraction = 45/360 = 1/8 Area = (1/8)(pi)(5^2) = (1/8)(pi)(25) ```

Answer 63

Composed of two circles on either end of a rolled up rectangle

Answer 64

Surface Area = 2 Circles + Rectangle = 2(Pi * r^2) + 2(Pi)(r)(h).

Answer 65

(1) radius of cylinder | (2) height of cylinder

Answer 66

V = (pi)(r^2)(h), where V is the volume, r is the radius of the cylinder, h is the height of the cylinder, and Pi is a constant that equals approximately 3.14. *NOTE: two cylinders can have the same volume but different shapes (and therefore each fits differently inside a larger object)

Answer 67

(1) radius of cylinder | (2) height of cylinder

Answer 68

Circle inscribed in a square is tangent to all sides Circle inscribed in a rectangle is tangent to two sides

Answer 69

``` In 8 minutes, or 480 seconds, 480(Pi) cm^3 of water flows into the tank. Therefore, the volume of the tank is 480(Pi). You are given the height of 80, so you can solve for the radius: V = (Pi)(r^2)(h) 480(Pi) = 30(Pi)(r^2) r^2 = 16 r = 4 ```

Answer 70

- Shortest distance between two points | - As an angle, a line measures 180 degrees

Answer 71

-Lines that lie in a plane and that never intersect no matter how far they are extended

Answer 72

Lines that intersect at a 90 degree angle

Answer 73

(1) Angles formed by any intersecting lines | (2) Angles formed by parallel lines cut by a transversal

Answer 74

(1) a + b + c + d = 360. Interior angles formed by intersecting lines form a circle, so the sum of these angles is 360 degrees (2) a + b = 180, b + c = 180, c + d = 180 and d + a = 180. Interior angles that combine to form a line sum to 180 degrees (supplementary angles) (3) a = c and b = d. Angles found opposite each other where these two lines intersect are equal (vertical angles) *These rules apply to more than two lines that intersect at a point

Answer 75

The exterior angle of a triangle is equal to the sum of the two non-adjacent (or opposite) interior angles a + b + c = 180 (sum of angles in a triangle) b + x = 180 (supplementary angles) Therefore, x = a + c *FREQUENTLY TESTED ON THE GMAT IN DISGUISED FORM

Answer 76

A line that passes through two lines in the same plane at two distinct points

Answer 77

The interior angles of two intersecting lines form a circle of 360 degrees

Answer 78

Vertical means that they share the same vertex (or corner point) Vertically opposite points are equal to each other

Answer 79

Two lines are parallel if one of the following is true: (1) Corresponding angles (angles in matching corners) are equal (a = e) (2) Alternate interior angles (on opposite sides of the transversal but inside the two lines) are equal (c = f) (3) Alternate exterior angles (on opposite sides of the transversal but outside the two lines) are equal (b = g) (4) Consecutive interior angles (on one side of the transversal but inside the two lines) add up to 180 (d + f = 180)

Answer 80

- 8 angles are formed but there are only TWO different measures - All acute angles (less than 90) are equal; and all obtuse angles (more than 90) are equal - Any acute angle is supplementary to any obtuse angle

Answer 81

The opposite angles are equal

Answer 82

The angles sum to 180 degrees

Answer 83

(x,y) - The values of a point on a number line - The first number in the pair is the x-coordinate, which corresponds to the horizontal location of the point as measured by the x-axis - The second number in the pair is the y-coordinate, which corresponds to the vertical location of the point as measured by the y-axis

Answer 84

The coordinate pair (0,0) represents the origin of a coordinate plane

Answer 85

A line can have one of four types of slope: (1) positive (2) negative (3) zero (4) undefined

Answer 86

m = Slope = Rise / Run = Change in y / Change in x - This tells you how steep the line is and whether the line is rising or falling - The numerator tells you how many units you want to move in the y direction (i.e. how far up or down you want to move) - The denominator tells you how many units you want to move in the x direction (i.e. how far left or right you want to move)

Answer 87

- Point on the line at which y = 0 | - To find x-intercepts, plug in 0 for y

Answer 88

- Point on the line at which x = 0 | - To find y-intercepts, plug 0 in for x

Answer 89

y = mx + b - m and b represent numbers (positive or negative) - m represents the slope of the line - b represents the y-intercept; it tells you where the line crosses the y-axis

Answer 90

- An equation that represents a straight line, often expressed as Ax + By = C, where A, B and C are constants - In all cases, a linear equation will never use terms such as x^2, sqrt(x), or xy

Answer 91

VERTICAL LINE If you know that x = 4, then the point can be anywhere along a vertical line that crosses the x-axis at (4,0)

Answer 92

HORIZONTAL LINE Every point two units below 0 fits this condition. These points form a horizontal line that crosses the y-axis at (0,-2). We do not know anything about the x coordinate

Answer 93

Every point to the right of the y axis has an x coordinate greater than 0. We do not know anything about y. Thus, the entire right-hand side of the y axis is shaded

Answer 94

The bottom right quadrant is shaded (quadrant IV). So you know that the point lies somewhere in the bottom right quarter of the coordinate plane

Answer 95

Use the Point-Slope Formula: Y – Y1 = m(X – X1) You can you use either point to replace the values of X1 and Y1 m = Change in Y / Change in X = (Ya – Yb) / (Xa – Xb) Slope-Intercept Formula: y = mx + b What are the steps for writing a linear equation with two given points? (1) Find the slope of the line by calculating rise over run (2) Plug the slope in for the m in the slop-intercept equation (3) Solve for b, the y-intercept, by plugging the coordinates of one point into the equation (4) Write the equation in the form y = mx + b

Answer 96

(1) Draw a right triangle connecting the points (2) Find the lengths of the two legs of the triangle by calculating the rise and the run (3) Use Pythagorean’s theorem to calculate the length of the diagonal, which is the distance between the points

Answer 97

- The four quarters of a coordinate plane are called quadrants - Each quadrant responds to a different combination of signs of x and y - The quadrants are always numbered starting on the top right and going counter-clockwise Q1: contains positive x values, positive y values Q2: contains negative x values, positive y values Q3: contains negative x values, negative y values Q4: contains positive x values, negative y values

Answer 98

Negative reciprocal. For example Line 2 is perpendicular to Line 1, and there slopes are: ``` m1 = 3 m2 = -1/3 ```

Answer 99

Slopes of parallel lines are identical

Answer 100

(1) side length (2) diagonal length (3) perimeter (4) Area

Answer 101

(1) radius (2) diameter (3) circumference (4) area

Answer 102

(1) one side length and its corresponding angle (2) perimeter (3) area

Answer 103

(1) DO NOT assume what you do not know for sure | (2) Use every piece of information you do have to make further inferences

Answer 104

(1) Draw or redraw figures (2) Fill in the given information (3) Identify the desired value (4) Infer from the givens using equations and relationships for unknowns (5) Solve the equations for the desired value

Answer 105

(1) Circle in Square: diameter = side length (2) Triangle in Rectangle: length + width of rectangle = base + height of triangle (3) Triangle in Trapezoid: height of trapezoid = height of triangle (4) Triangle in Circle: radii of circle = sides of triangle (5) Square in Circle = diameter of circle = diagonal of square

Answer 106

The triangle must be a right triangle, with the right angle lying opposite a semicircle Conversely, any right triangle inscribed in a circle must have the diameter of the circle as one of its sides

Answer 107

- Those that can change size but never change shape (e.g. squares, equilaterals, circles, spheres, cubes, 45-45-90 triangle, 30-60-90 triangles, and others) - You only need ONE measurement in order to know EVERY measurement - If you have two regular figures, and you know how they are related numerically (e.g. the side of an equilateral triangle has the same length as the diagonal of a square), then you can conclude that ANY measurement for EITHER figure will give you ANY measurement for either figure

Answer 108

Step 1: when geometry questions involve a figure, redraw it Step 2: fill in the given information Step 3: Identify the wanted element Step 4: Infer from the givens using variables you already have BAC + 3x = 180 --> BAC = 180 – 3x BCA + 2x = 180 --> BCA = 180 – 2x 40 + (180 – 3x) + (180 – 2x) = 180 400 – 5x = 180 220 = 5x x = 44

Answer 109

Break Quadrilateral ABCE into two pieces: a 3 by 2x rectangle and a right triangle with a base of x and a height of 3. Therefore, the area of Quadrilateral ABCE is given by the following: (3*3x) + (3x)/2 = 9x + 1.5x = 10.5x If ABCE has an area of 21, then 21 = 10.5x and x = 2. Quadrilateral is a parallelogram; thus, its area is equal to base*height or 4x*3. A = 4(2)(3) = 24

Answer 110

Area of Circle is 72(Pi) – 72 square units larger than the area of triangle ABC ``` Triangle ABC is a 45-45-90 triangle, and the base and height are equal. Assign the variable x to represent both the base and height: A = bh/2 72 = (1/2)(x^2) 144 = x^2 x = 12 ``` The hypotenuse is equal to 12√2. Therefore, the radius is equal to 6√2 and the area of the circle is 72(Pi)

Answer 111

Answer D. These triangles are similar. Any time two triangles each have a right angle and also share an additional right angle (or, in this case, the 90 degree span on either side of Point C, they will be similar). Since the two triangles are right triangles, if you had any two sides of triangle ABC, you could get the third. Because the two triangles are similar, you could use any two sides of triangle EDC (note that you already have DE = 16), as well as the ratio of one triangle’s size to the other, to get the third side of EDC as well as all three sides of ABC. (1) SUFFICIENT. Side DC = 20. Use the 20 and the 16 to get, via the Pythagorean theorem, that side CE = 12. Then AC = 8. Thus, you have all three sides of EDC, plus the ratio of one triangle to the other (side AC = 8 matches up with side DE = 16; thus the smaller triangle is one-half the size of the larger) (2) SUFFICIENT. Side AC = 8. Note that this gives you the same information as statement 1. If AC = 8, then CE = 12, and you can calculate all three sides of EDC

Answer 112

(1) PS: “what is the maximum area of…?” | (2) DS: “is the area of rectangle ABCD less than 30?”

Answer 113

- Of all quadrilaterals with a given perimeter, the square has the largest area (even in cases involving non-integer lengths) - Of all quadrilaterals with a given area, the square has the minimum perimeter *Both of these principles can be generalized for n sides: a regular polygon with all sides equal, will maximize area for a given perimeter and minimize perimeter for a given area

Answer 114

- If you are given two sides of a triangle or parallelogram, you can maximize the area by placing those two sides perpendicular to each other * Maximize area of rhombus by turning it into a square

Answer 115

- Any function in the form of f(x) = ax^2 + bx + c, where a, b and c are constants - Can be plotted as a parabola in the coordinate plane Positive value for a = curve opens upwards Negative value for a = curve opens downwards Large ǀaǀ = narrow curve Small ǀaǀ = wide curve

Answer 116

(1) solve for zero by factoring and solving the equation directly (2) plug in points and draw the parabola (3) use the quadratic formula x = [-b ± √(b^2 – 4ac)] / 2a The discriminant, b^2 – 4ac, tells you how many solutions exist - Positive: two roots - Zero: one root - Negative: zero roots

Answer 117

- The graph of a quadratic function - A quadratic function is an equation containing exponents or roots - For example, y = x^2 is a quadratic function

Answer 118

A line or line segment that cuts a line segment exactly in half.

Answer 119

- A line or line segment that cuts a line segment exactly in half and forms a 90 degree angle with that line - Perpendicular bisector has the negative reciprocal slope of the line segment it bisects (ie the product of the two slops is -1; exceptions are vertical and horizontal lines)

Answer 120

(1) Find the slope of AB. m = (y1 – y2) / (x1 – x2) = (2 + 2) / (2 – 0) = 4/2 = 2 (2) Find the slope of the perpendicular bisector. m = negative reciprocal = -1/2 (3) Find the midpoint of AB. x coordinate will be the average of the x coordinates for points A and B and the y coordinate will be the average of the y coordinate for points A and B A: (2,2) Midpoint: (1,0) B: (0, -2) (4) Put the information together. To find the value of b, the y intercept, substitute the coordinates of midpoint for x and y 0 = -1/2(1) + b b = 1/2 The perpendicular bisector of segment AB has the equation y = (-1/2)x + ½

Answer 121

Slope = negative reciprocal = -1/m

Answer 122

- The average of the x coordinates for points A and B - The average of the y coordinates for points A and B [(x1 + x2)/2 , (y1+y2)/2]

Answer 123

``` 2x + 3(4x – 10) = 26 2x + 12x – 30 = 26 14x = 56 X = 4 Y = 6 ``` Takeaway: - If two lines in a plane do not intersect, then the lines are parallel (ie there is no pair of numbers (x,y) that satisfies both equations at the same time - Two linear equations might represent the same line in which case, infinitely many points (x,y) along the line satisfy the two equations

Answer 124

A right triangle is half of a rectangle. Constructing a right triangle with legs that add up to 40 is equivalent to constructing a rectangle with a perimeter of 80. Maximize the area of the rectangle. Each side should have length of 80/4 = 20. Thus, the maximum area of the rectangle is 400, and the maximum area of the triangle is one-half of that value, or 200 square units.

Answer 125

You can calculate the area of the solid triangle as (1/2)(12)(3) = 18 Next, use the side length of 7 as the base and write the equation for the area as (1/2)(7)(x) ``` 18 = (1/2)(7)(x) 7x = 36 x = 36/7 ```

Answer 126

The line segment AB must have a slope of 0.5 (negative reciprocal of -2) ``` Y = 0.5x + b 2 = 0.5(7) + b b = -1.5 ``` The line containing AB is y = 0.5x – 1.5. Find the point at which the perpendicular bisector intersects AB by setting the two equations equal to each other -2x + 6 = 0.5 – 1.5 2.5x = 7.5 X = 3; Y = 0 The two lines intersect at (3,0), which is the midpoint of AB Point B has coordinates (-1, -2)

Answer 127

We have two similar triangles, such that the smaller one is equal to 1/4 of the larger one Find the x coordinate: 4x = 3 --> x = 0.75 --> x-coordinate = -2.75 Find the y coordinate: 4x = 6 --> x = 1.50 --> y-coordinate = 1.50 (-2.75, 1.50) is the point that is three times as far from A as from B and that is between points A and B

Answer 128

We know the perimeter of the rectangle is: 2W + 2L = 10 W + L = 5 We also know the area of the rectangle is: (W)(L) = 6 ``` Use substitution to solve for one side: W = 5 – L (5 – L)(L) = 6 5L – L^2 = 6 L^2 – 5L + 6 = 0 (L – 3)(L – 2) = 0 L = 3 OR L = 2 ``` Either the length is 3 and the width is 2, or the length is 2 and the width is 3

Answer 129

If the square has an area of 49, then (side)^2 = 49. That means that the length of the sides of the square is 7. Now we can use the Pythagorean Theorem to find the length of the diagonal line, which is also the hypotenuse 7^2 + 7^2 = c^2 c = Sqrt(98) = Sqrt(2 * 7 * 7) = 7Sqrt(2)

Answer 130

of diagonals = n(n - 3) / 2 where n is the number of sides of the shape

Answer 131

(i) Then the areas of the triangles will have the ratio of (1/2^2) = 1/4 (ii) Then the areas of the triangles will have the ratio 2^2 = 4

Answer 132

SSS: if two triangles have three congruent sides, then the triangles are congruent SAS: if two triangles have two congruent sides and the angles formed by these sides are congruent, then the triangles are congruent ASA: if two triangles have two congruent angles and the adjacent sides are also congruent, then the triangles are congruent AAS: if two triangles have two congruent angles and one congruent side, then the triangles are congruent

Answer 133

6 equilateral triangles

Answer 134

f(x) = ax^2 + bx + c, (1) x^2 + 3 ---- graph moves up three units on y axis (2) (x + 3)^2 ---- graph moves to the left three units on x axis (3) Positive value for a = curve opens upwards (4) Negative value for a = curve opens downwards (5) Large ǀaǀ = narrow curve (6) Small ǀaǀ = wide curve