Special Triangles

Video #706: Special Triangles

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


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

A very important point about equilateral triangles that you should never forget it that when you draw an altitude, you have not simply created two congruent triangles.  They are 30:60:90 triangles.  Look at them on the “cheat sheet.”  The sides are always in the same ration that you see there.  This is one way the test writers tell you that a triangle is a 30:60:90; they split an equilateral triangle in half.


In the isosceles triangle at the left, 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

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? Answer #12

Similar Triangles:

Words of “Wiz-dom”-Similar triangles come in all shapes and sizes. However, they are “special.” They have a partner that has exactly the same sized angles BUT have sizes of different lengths. (If the angles AND sides are equal the triangles are congruent.) The test writer likes to ask questions about similar triangles that are really proportion questions. Just set up the lengths of the corresponding sides (using the same-sized angles as your guide) of the similar triangles in a proportion. The ratio of the lengths of the corresponding sides will remain constant between the two triangles.

The “Cheat Sheet”:

Words of “Wiz-dom”-Get used to using the geometry information provided at the beginning of the math section of the SAT. It is the best “cheat sheet” you can use on the SAT! Some of the material that is reviewed above is mentioned there. Some additional important information also can be found there. Most importantly you’ll see the ratios of the sides of 30:60:90 and 45:45:90 triangles. Keep your text open to a page that has this information while you do the sample questions below.

Video #706: Special Triangles

Sample Questions

8.  What is the perimeter of the triangle above?

(A) 3
(B) 6
(C) 9
(D) 25
(E) 27

Answer to Question #8

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

(A) 12
(B) 16
(C) 32
(D) 48
(E) 64

Answer to Question #9

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


Answer to Question #6

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

(A) 24
(B) 32
(C) 48
(D) 64
(E) 96

Answer to Question #7

10.  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
(E) 6

Answer to Question #10

Video #706: Special Triangles

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