# Special Triangles

Words of “Wiz-dom”—There are three kinds of triangles that are important on the ACT: equilateral, isosceles, and right. Let’s do a review of these special triangles before we look at some sample questions.

Equilateral:

In triangle QRS, what are the sizes of angles x, y and z? Answer #1

If QR has a length of 3, what is the length of QS? Answer #2

Since QR is 3, what is the ratio of QR:RS:QS? Answer #3

Which one of the basic triangle rules is best exemplified by an equilateral triangle? Answer #4

If QR and QS don’t change lengths and angle x is reduced in size, what would happen to side RS? angle y? angle z? Answer #5

### Isosceles

In the isosceles triangle above, DE is 10 and angle a is 20 degrees.

What is the length of EF? Answer #6

What is the size of angle b? Answer #7

What is the size of angle c? Answer #8

If angle a is increased 20 degrees and the resulting triangle is an isosceles triangle, what happens to angle b? Answer #9

### Right Triangles

How many degrees are at T? The sum of U and V? Answer #10

Which side of a right triangle has to be the longest? Why? What is it called? Answer #11

Words of “Wiz-dom”-The two right triangles are 30:60:90 and 45:45:90. Notice the ratios of the size of the angles are 1:2:3 and 1:1:2 respectively. the ratios of the size of the sides is

The Pythagorean Theorem states a2 + b2 = c2.

Which side is a? b? c? If a equals 2 and b equals 4, how long is c?

Words of “Wiz-dom”—Remember the Pythagorean Theorem triplets 3:4:5 and 5:12:13. In addition, there are two special right triangles that you need to understand and memorize for the ACT. (Unlike the SAT, you have to memorize all math formulas, relationships, and conversion factors for the ACT. The SAT has a “cheat sheet” (called the “Reference Information” by the SAT test writers) that includes everything you need to know. ACT test takers need to memorize everything.) Memorize the ratios of the sides of 30:60:90 triangles and 45:45:90 triangles . Also keep in mind that anytime you draw an altitude of any triangle you are creating two right triangles. (Pythagoras usually becomes your best friend when you draw that line!)

## Similar Triangles

Words of “Wiz-dom”—One of the test writer’s favorite items is similar triangles. They are triangles in which the angles are all equal and the sides are unequal but in proportion to corresponding sides on each other. These questions are really just ratio problems in the form of a geometry item. Usually, the test writer will tell you that two triangles are similar triangles, give you the lengths of 2 corresponding sides and ask you to calculate the length of another corresponding pair of sides.

Sometimes they sneak a geometric principle about triangles onto the test. In any large triangle that has a line drawn inside it that is parallel to one of its sides, the smaller triangle is similar to the larger one. For example:

If sides AB and CD are parallel, the small triangle is similar to the large one. The ratio of lengths of the corresponding sides will be the same. That means AB:CD::AE:AC::BE:BD. That means if AE is twice as long as AC then AB is twice as long as CD and BE is twice as long as BD. What does it mean for the area of the small triangle compared to the large one?

## Sample Questions

1.  What is the perimeter of the triangle above?

A. 3
B. 6
C. 9
D. 25
E. 27

2.  If AB equals BC, what is the area of ABC?

A. 12
B. 16
C. 32
D. 48
E. 64

3.  If the legs of a right triangle are of lengths 2f and 3g respectively, how long is the hypotenuse?

A. 6fg
B.
C.
D.
E.

4.  ABC has a perimeter of 36. What is the area of the triangle?

A. 24
B. 32
C. 48
D. 64
E. 96

5.  The sizes of the angles in the triangle above have a ratio of 1:2:3. The shortest side has a length of 2. What is the perimeter of the triangle?

A. 16
B. 12
C.
D.
E. 6