Types of Triangles
You can classify triangles into three categories based on the number of equal sides.
■Scalene Triangle: no equal sides
■Isosceles Triangle: two equal sides
■Equilateral Triangle: all equal sides
You also can classify triangles into three categories based on the measure of the greatest angle:
■Acute Triangle: greatest angle is acute
50° 60°
70°
Acute
Equilateral
Isosceles
Scalene
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■Right Triangle: greatest angle is 90°
■Obtuse Triangle: greatest angle is obtuse
Angle-Side Relationships
Understanding the angle-side relationships in isosceles, equilateral, and right triangles is essential in solving ques-
tions on the SAT.
■In isosceles triangles, equal angles are opposite equal sides.
■In equilateral triangles, all sides are equal and all angles are 60°.
60º
60º60º
ss
s
22
m∠a = m∠b
130°
Obtuse
Right
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■In right triangles, the side opposite the right angle is called the hypotenuse.
Practice Question
Which of the following best describes the triangle above?
a. scalene and obtuse
b. scalene and acute
c. isosceles and right
d. isosceles and obtuse
e. isosceles and acute
Answer
d. The triangle has an angle greater than 90°, which makes it obtuse. Also, the triangle has two equal sides,
which makes it isosceles.
Pythagorean Theorem
The Pythagorean theorem is an important tool for working with right triangles. It states:
a2b2c2,where aand brepresent the lengths of the legs and crepresents the length of the hypotenuse of a
right triangle.
Therefore, if you know the lengths of two sides of a right triangle, you can use the Pythagorean theorem to
determine the length of the third side.
40°
100°
40°
66
Hypotenuse
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Example
a2b2c2
3242c2
9 16 c2
25 c2
25
c2
5 c
Example
a2b2c2
a262122
a236 144
a236 36 144 36
a2108
a2
108
a108
a
12
6
4c
3
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Practice Question
What is the length of the hypotenuse in the triangle above?
a. 11
b. 8
c. 65
d. 11
e. 65
Answer
c. Use the Pythagorean theorem: a2b2c2,where a7 and b4.
a2b2c2
7242c2
49 16 c2
65 c2
65
c2
65
c
Pythagorean Triples
A Pythagorean triple is a set of three positive integers that satisfies the Pythagorean theorem, a2b2c2.
Example
The set 3:4:5 is a Pythagorean triple because:
324252
9 16 25
25 25
Multiples of Pythagorean triples are also Pythagorean triples.
Example
Because set 3:4:5 is a Pythagorean triple, 6:8:10 is also a Pythagorean triple:
6282102
36 64 100
100 100
7
4
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