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GMAT Quant > GMAT Quant Chapter 17: Geometry > Flashcards

GMAT Quant Chapter 17: Geometry Flashcards

1
Q

Define supplementary angles.

A

Their angles sum to 180 degrees.

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2
Q

What is the relationship between the size of the sides and the size of the angles in a triangle?

A

The largest angle is always opposite the longest side of the triangle, the smallest angle is always opposite the shortest side of the triangle.

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3
Q

define an exterior angle

A

the angle created by one side of the triangle and the extension of an adjacent side

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4
Q

What is the sum of the exterior angles for any polygon?

(taking one exterior angle in a straight line from each vertex)

A

360 degrees

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5
Q

What fact do we know about the interior and exterior angles of a triangle?

A

Exterior angle is equal to the sum of the two interior angles it is not connected to

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6
Q

What is the standard formula for the area of a triangle?

A

Base x Height x 1/2

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7
Q

What inequality is there about the length of a triangle’s sides?

A

The sum of 2 lengths of any 2 sides of the triangle > length of the third side

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8
Q

Define isosceles triangle

A

At least 2 sides of the same length, at least two angles are the same

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9
Q

What are the 3 most common Pythagorean Triples?

A

3-4-5
5-12-13
8-15-17

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10
Q

What is another name for an isosceles right triangle and what are its properties?

A

45-45-90 triangle
Area = equal length sides ^2 / 2
ratio of the sides = x : x : x Root 2
Area is half of the square

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11
Q

What is the area of an equilateral triangle?

What kind of triangle appears when we cut an equilateral triangle in half?

A

side ^ 2 Root 3 / 4

2 x 30-60-90 triangles

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12
Q

What are the 3 versions of triangle similarity (scale)?

A
  1. 3 angles measure the same for both triangles
  2. 3 pairs of corresponding sides have lengths in same ratio
  3. Angle of 1 triangle = angle of another triangle and the sides are in the same ratio
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13
Q

What are laws about the squares and rectangles?

A

All squares are rectangles
All rectangles are parallelograms

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14
Q

What facts do we know about parallelograms?

A

opposite sides are equal in length
opposite angles are equal in measure
any 2 consecutive angles are supplementary

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15
Q

What must we remember about the division of a rectangle into two right triangles?

A

only divide into two 30-60-90 triangles if the ratio of width : length is…

X: X Root 3
or
1 : Root 3

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16
Q

What is the length of the diagonal line between two corners of a square?

A

side Root 2

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17
Q

What fact do we know about the maximum area of a rectangle?

What fact do we know about the minimum perimeter of a rectangle?

A

If a square and a rectangle have the same perimeter, the square will always have the greater area.

If the rectangle and square have equal areas, the square will always have the smallest perimeter.

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18
Q

What formula do we use to calculate the area of a trapezoid?

What do you call a trapezoid with two parallel sides of equal length?

A

Area = (base 1 + base 2) * height / 2

isosceles trapezoid

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19
Q

How can we calculate the sum of the interior angles of any polygon?

How can we calculate the interior angle of a regular polygon?

A

(n -2) * 180

where n is the number of sides of the polygon

(n-2) * 180 / n

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20
Q

Define “regular” when it is used in the phrase regular polygon.

A

All sides are the same length and all interior angles are equal

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21
Q

What is the formula for the area of a regular hexagon?

What is the alternative method for calculating the area of a regular hexagon?

A

3 Root 3
/
2 side ^2

Area = 1.5ds
D is the distance between 2 parallel sides
S is the length of the sides

22
Q

What fact do we know about a regular hexagon?

A

It can be divided into 3 equilateral triangles by drawing 3 diagonals

23
Q

What is the sum of the exterior angles for any polygon?

A

360 degrees

24
Q

What is the longest chord that can be drawn in a circle?

A

The diameter.

25
Q

On the GMAT, if we need to approximate pie what value should we use?

A

3

26
Q

What 3 equations of equivalence are there concerning a circle?

A

Central angle / 360

any arc length/circumference
=
area of sector / area of circle

27
Q

What is the relationship between a central angle in a circle and an inscribed angle in a circle when they both share the same endpoints?

A

The central angle is double the inscribed angle.

28
Q

What is the relationship of the size of an inscribed angle and the angle of an arc that its two intercepts create?

A

the inscribed angle is 1/2 of the arc’s angle

29
Q

For a triangle inscribed in a circle, what must be true if one of the triangle’s sides is the diameter of the circle?

A

The triangle is a right triangle.

30
Q

If we are given a right triangle inscribed in a circle, where can we draw 1 line to work out the length of the arc?

A

Draw a line from the right angle of the triangle to the centre of the circle.

This creates two radii of the same length which create the arc. Solve using the triple equivalence formula.

31
Q

What formula can we use to calculate the area of an equilateral triangle?

A

s^2 Root 3
/
4

32
Q

What fact do we know must be true when an equilateral triangle is inscribed in a circle?

What two deductions can we make from this?

A

The centre of the triangle and the centre of the circle are at the same point.

A line from the vertex of the triangle to the center of the circle is the radius of the circle.

A line from the center of the triangle to the base of the triangle creates a 30-60-90 triangle.

33
Q

An equilateral triangle inscribed in a circle breaks the circle up into how many arcs of equal length?

When a circle is inscribed in an equilateral triangle, what do we know about the points of the circle that touch the triangle?

A

3

The points of a circle inscribed in an equilateral triangle where the circle and triangle touch are the midpoints of the sides of the triangle.

34
Q

When a square is inscribed inside a circle what is the diagonal line between two vertexes of the square equivalent to?

When a rectangle is inscribed in a circle, what is the diagonal line between its two vertexes equivalent to?

A

The diameter of the circle.

The diameter of the circle.

35
Q

For a circle inscribed in the square:

  1. what is the largest circle that can fit in the square?
  2. which points are the midpoints of the square’s sides?
  3. What part of the circle is equivalent to the side of the square?
A
  1. when each side of the square is a tangent of the circle, the circle is the largest one that can fit the square
  2. the 4 places the circle touches the square
  3. The diameter.
36
Q

The longest side of a triangle inscribed in a square is equivalent in length to which part of the square?

A

Longest side of the triangle inscribed in a square is equivalent to the side of the square

37
Q

The two right triangles formed by the diagonal of a rectangle inscribed in a circle only become 30-60-90 right triangles if the ratio of the sides of the rectangle are?

A

X to X root 3

38
Q

When a regular polygon is inscribed in a circle, how many parts of the circle’s circumference are divided into equal lengths?

A

The circumference of the circle is divided into the same number of equal lengths as the to the number of sides of the polygon that is inscribed within it.

39
Q

A square inscribed within a square:

  • Where does the vertex of the smaller square lie when the smaller square has the smallest area possible?
  • How can we make the area of the inscribed circle larger?
A

At the midpoints of the larger square’s sides

Move the vertex’s of the smaller square closer towards the vertex’s of the larger square.

40
Q

What useful shape can we create when a square is inscribed within a semicircle?

A

A right triangle where the radius of the semi-circle is the hypotenuse, the side of the square is one leg and half the length of the square is the other leg

41
Q

When there is a uniform border around a rectangle, what are its new dimensions?

Explain how many times we add the uniform border measurement to figure out the new dimensions of the larger shape in words.

A

Width = Width + 2x
Length = Length + 2x

Where x is the length of the uniform border

We add the distance of the uniform border twice to each dimension (length and width) in order to know the dimensions of the new shape including the border because the border is added to both sides of the shape.

42
Q

How do we calculate the area of the uniform border around a rectangle?

A

(Length + 2x)(Width + 2x) - WL

43
Q

How do we calculate the area of the outside circular ring?

A

pie (R1^2 - R2^2)

R1 is the radius of the largest circle
R2 is the radius of the smallest circle

44
Q

What is the volume of a cube?

A

Side^3

45
Q

How can figure out the volume of a rectangular solid without knowing the width, length, and height individually?

A

The surface area of one side of the shape x the other unknown dimension

46
Q
  1. How do we calculate the longest line segment that can be drawn within a rectangular solid?
  2. What is the formula for the diagonal of the cube?
A

1.The longest line segment is from one vertex through the middle to the opposite vertex.

Use Pythagorean theorem to calculate

  1. Side Root 3
47
Q

What is the surface area of a cube?

What is the surface area of a rectangular object?

A

6 side ^2

2 (LW) + 2(HW) + 2(LH)

48
Q

What is the volume of a right circular cylinder?

A

pie * r^2 * h

49
Q

What is the surface area of a right cylinder?

How can we visualize the surface area of a right cylinder?

A

2(pier^2) + 2(pier*h)

There are 3 parts. 2 parts are the circle at the top and bottom. The third part is the rectangle that is wrapped around shape.

50
Q

In order to calculate the rate at which an object fills with liquid, what specific information must we know in order to answer correctly?

To figure out the number of objects that fit inside a particular space, what must we know?

A

The exact dimensions of the shape and the rate at which the liquid flows into the shape.

The exact dimensions of both shapes.

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