Algebal review 4 pot

Types of TrianglesYou can classify triangles into three categories based on the number of equal sides.. ■ Scalene Triangle: no equal sides ■ Isosceles Triangle: two equal sides ■ Equilat

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

70°

Acute

Equilateral Isosceles Scalene

■ 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

m∠a = m∠b

130°

Obtuse Right

■ 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.

 P y t h a g o r e a n T h e o r e m

The Pythagorean theorem is an important tool for working with right triangles It states:

a2 b2 c2, where a and b represent the lengths of the legs and c represents 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°

Hypotenu

se

a2 b2 c2

32 42 c2

9  16  c2

25  c2

25  c2

5  c

Example

a2 b2 c2

a2 62 122

a2 36  144

a2 36  36  144  36

a2 108

a  1082 

a 108

6

3

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: a2 b2 c2, where a  7 and b  4.

a2 b2 c2

72 42 c2

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, a2 b2 c2

Example

The set 3:4:5 is a Pythagorean triple because:

32 42 52

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:

62 82 102

36  64  100

100  100

7

4

Pythagorean triples are important because they help you identify right triangles and identify the lengths of the sides of right triangles

Example

What is the measure of∠a in the triangle below?

Because this triangle shows a Pythagorean triple (3:4:5), you know it is a right triangle Therefore,∠a must

measure 90°

Example

A right triangle has a leg of 8 and a hypotenuse of 10 What is the length of the other leg?

Because this triangle is a right triangle, you know its measurements obey the Pythagorean theorem You could plug 8 and 10 into the formula and solve for the missing leg, but you don’t have to The triangle shows two parts

of a Pythagorean triple (?:8:10), so you know that the missing leg must complete the triple Therefore, the sec-ond leg has a length of 6

It is useful to memorize a few of the smallest Pythagorean triples:

3:4:5 3 2 + 4 2 = 5 2

6:8:10 6 2 + 8 2 = 10 2

5:12:13 5 2 + 12 2 = 13 2

8 10

?

a

4