Chapter 2- Traingles & DIagonals

Question 1

List 2 key properties in any given triangle?

Answer 1

1) The sum of the three angles of a triangle equals 180.
2) Angles correspond to their opposite sides.

Question 2

What is the triangle inequality law

Answer 2

the sum of any two sides of triangle must be greater than the third side ( Note: the sum of the two sides cannot be equal to the third side, it must be GREATER than the third) AND The third side must be greater than the difference between the lengths of the other two sides.

** 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.**

Question 3

List the common combinations of common right triangle:

3 - 4 - 5

Answer 3

3 - 4 - 5 (The most popular of all right triangles)

3²+ 4²= 5² = (9 + 16 = 25)

Key Multiples:

2* (6-8-10)

3* (9-12-15)

4* (12-16-20)

Question 4

List the common combinations of common right triangle:

5-12-13

Answer 4

Also quite popular on the GMAT

5²+12²=13²= (25+144=169)

Key Multiples:

*2 (10-24-26)

Question 5

List the common combinations of common right triangle:

8-15-17

Answer 5

Not as frequent on the exam

8²+15²=17² = (64+225=289)

No common combination

Question 6

What is the ratio of the isosceles triangle (most important isosceles triangle on the GMAT is the isosceles right triangle)

Answer 6

The isosceles right triangle has one 90 angle ( Opp hypo) and two 45 angles ( opp the two equal legs).

The lengths of the legs of every 45-45-90 triangle have a specific ratio:

45 45 90

leg leg hypotenuse

1 : 1 : √2

x : x: x√2

Question 7

Why is the isosceles triangle (45-45-90) so imp!

Answer 7

Isosceles triangle is very important because it is exactly half of a square. So if you put 2 (45-45-90) triangles together, you get a square.

Therefore, if you are given the diagonal of the square, you can use the 45-45-90 ratio to find out the length of a side of the square.

Question 8

What is the ratio of the equilateral triangle

Answer 8

Equilateral triangle is made up of 2, 30-60-90 triangles.

30 60 90

short leg long leg hypotenuse

1 : √3 : 2

x : x√3 : 2x

Question 9

Given a square with a side of length 5, what is the length of the diagonal of the square?

Answer 9

d= diagonal of square

d= s√2, where s is a side of the square

Ans: length of the diagonal of the square 5 √2 (5 is s (side)

Question 10

What is the measure of an edge of a cube with a main diagonal of length √60?

Answer 10

diagonal (d) of cube d = s√3, where s is a side of the square

Ans: length of a diagonal of the cube is d = s√3 √60=s√3 s=√60/√3= s=√20

Question 11

If the rectangle has a length of 12 and a width of 5, what is the length of the diagonal?

Answer 11

To find the diagonal of a rectangle, you must know EITHER the length and the width OR ONE dimension and the proportion of one of the other.

Ans: 5-12-x is the common right angle. x = 13

Question 12

What is the formula for the “Deluxe” Pythagorean Theorem

Answer 12

d²= x²+y²+z²

(x,y,z are the sides of the rectangular solid and d is the main diagonal).

This formula will provide you with main diagonal of a rectangular solid ( 3 D shape)

Question 13

Define similar triangles?

Answer 13

Triangles are defined as similar if all their corresponding angles are EQUAL and their CORRESPONDING SIDES ARE IN PROPORTION.

**IMP - Any time two triangles EACH have a right angle and ALSO share an additional right angle ( or a 90 degree span in the middle to the two right triangles), they will be similar.

Question 14

Describe an observation that can be generalized about similar triangles in terms of its ratio?

Answer 14

If two similar triangles have corresponding side lengths in ratio a:b, then their areas will be in ratio a²:b². This principle is not limited to only triangles.

This principle holds true for ANY similar figures: quadrilateral, pentagon etc.

For similar solids with corresponding sides in ratio a:b, their volumes will be in ratio a³: b³.