Ch. 2_Triangles & Diagonals

The lengths of the legs of every 45-45-90 triangle have a specific ratio: MEMORIZE 1. Leg:Leg:hypotenuse

1:1:√2

The lengths of the legs of every 30-60-90 triangle have the following ratio: MEMORIZE short leg:long leg:hypotenuse

1:√3:2 x:x√3:2x

A close relative of the equilateral triangle is

30-60-90 triangle. These two triangles, put together, form an equilateral triangle.

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

Using the formula d = s√2, we find the length of the diagonal of the square is 5√2.

*Right* triangles have three possible bases

but they are special because their two legs are perpendicular. Therefore, if one of the legs is chosen as the base, then the other leg is the height.

3 Common right triangles MEMORIZE

1. 3-4-5 2. 5-12-13 3. 8-15-17

The sum of three angles of a triangle =

180º

2. 5-12-13

5^2 + 12^2 = 13^2 Key multiple: 10-24-26

The lengths of the legs of every 30-60-90 triangle have the following ratio: MEMORIZE short leg:long leg:hypotenuse Example: Given that the short leg of a 30-60-90 triangle has a length of 6, what are the lengths of the long leg and hypotenuse?

=6:6√3:2(6) =6:6√3:12

Area of right triangle =

A = 1/2(one leg) x (other leg) = 1/2(hypotenuse x height from hypotenuse)

The most important isosceles triangle on the GMAT

Isosceles *right* triangle

3. 8-15-17

Less frequent. There are no key multiples.

Angles corresponds to their opposite sides

This means that the largest angle is opposite the longest side, while the smallest angle is opposite the shortest side.

The polygon most commonly tested

Triangle

Isosceles triangles

Two sides are equal. The two angles opposite those two sides will also be equal.

The 45-45-90 right triangle is exactly half of

a square Two 45-45-90 triangles put together make up a square.

Pythagorean Theorem =

a^2 + b^2 = c^2 *applies to right triangles.

Triangles are defined as similar if

all the corresponding angles are *equal* and their corresponding sides are *in proportion.*

The diagonal of a square can be found by using this formula:

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

The main diagonal of a cube can be found using this formula:

d = s√3, where s is an edge of the cube.

The sum of two sides cannot be

equal to the third side.

If two sides are equal, their

opposite angles are also equal. Also called *isosceles* triangle.

If two similar triangles have corresponding side lengths in ratio a:b, then

their areas will be in ratio a^2:b^2

*Watch out for imposter triangles.

A random triangle with sides equal to 3 and 4 does not necessarily have a third side equal to 5.

1. 3-4-5

Most popular of all right triangles. 3^2 + 4^2 = 5^2 Key multiples: 6-8-10 9-12-15 12-16-20

Triangles that require attention

Right triangles, because they have special properties that are useful for solving many GMAT geometry problems.

The lengths of the legs of every 45-45-90 triangle have a specific ratio: MEMORIZE 1. Leg:Leg:hypotenuse Example: Given that the length of side AB is 5, what are the lengths of sides BC and AC?

Since AB is 5, we use the ratio: 1:1:√2 for sides AB:BC:AC to determine that the multiplier x is 5. We then find that the sides of the triangles have lengths 5:5:5√2. Therefore, the length of side BC = 5 and the length of side AC = 5√2.

Deluxe Pythagorean Theorem =

d^2 = x^2 + y^2 + z^2, where x, y, and z are the sides of the rectangular solid and d is the *main* diagonal.

Right triangles are helpful for

finding the diagonals of other polygons, specifically squares, cubes, rectangular solids.

The sum of any two sides of a triangle must be

greater than the third side.

The sum of two sides must be

greater than the third side.

Equilateral triangles

have three sides and three angles that are all equal. Thus, each angle of an equilateral angle is 60 since all three must sum to 180.

Right triangles are essential for solving problems

involving other polygons.

Since a triangle only has one area,

the area must be the same regardless of the side chosen as the base.

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.

The lengths of the legs of every 45-45-90 triangle have a specific ratio: MEMORIZE 1. Leg:Leg:hypotenuse Thus,

x:x:x√2

Given the lengths of any two sides of a *right* triangle,

you can determine the length of the third side using the Pythagorean Theorem.

Once you find that 2 triangles have 2 pairs of equal angles,

you know that the triangles are similar.

To find the diagonal of a rectangle

you must know *either* the length and the width OR one dimension and the proportion of one to the other.

There are two special types of right triangles:

1. 30-60-90 triangle 2. 45-45-90 triangle *You only need the length of ONE side to determine the lengths of the other two sides.*

Every right triangle is composed of

1. Two legs (often called a and b; it does not matter which side is a or b) 2. Hypotenuse (opposite side of the right triangle and is often the letter c)

Area of an equilateral triangle with a side of length S is =

1/2(S)(S√3/2) = (S^2√3)/4 The equilateral triangle has base length S and a height of length S√3/2