Mrs. McConaughy Geometry 1

Lesson 7.3 Two Special Right Triangles

Objectives: To use properties of 45-45-90

trianglesTo use properties of 30-60-90

triangles

Mrs. McConaughy Geometry 2

Isosceles Right Triangle Theorem

ISOSCELES RIGHT TRIANGLE THEOREM: In an isosceles right triangle, if the legs have length, l, then the hypotenuse has length ____.

l √2

NOTE: If you are given the length of

the hypotenuse, you can determine the

length of a side by dividing it by_________________________

___________________________.

√2, then rationalizing the

denominator, when necessary.

Mrs. McConaughy Geometry 3

EXAMPLES: Find the length of the hypotenuse in each isosceles triangle below.

3√2 4√2 5√2

6√2 7√2 12√2

Mrs. McConaughy Geometry 4

Recall: Triangle Inequalities

If two angles of a triangle are not congruent, then the longest side lies opposite the _______ angle and the shortest side lies opposite the ________ angle.

smallest

largest

Mrs. McConaughy Geometry 5

30-60-90 TRIANGLE THEOREM

30-60-90 TRIANGLE THEOREM: In a 30-60-90 triangle, if the side opposite the 30 degree angle has length, l, the hypotenuse has length_______.

NOTE: These triangles are sometimes referred to as 1-2-√3 right triangles.

2l

Mrs. McConaughy Geometry 6

Easy way to remember the relationship among angles and sides in

30-60-90 triangles: 1. Rank order the following numbers from smallest to largest:

1, 2, √3

2. Now, use the Triangle Inequality Theorem to place the side lengths 1l, √3l , 2l opposite the appropriate angles in a

30-60-90 triangle.

1, √3 , 2

30

60

1l2l

l√3

NOTE: It is usually easier to determine the length of the shortest and longest sides, initially.

Mrs. McConaughy Geometry 7

Find the length of each indicated side:

30

60

____

____

____NOTE: The length of one side will be provided by your instructor.

Mrs. McConaughy Geometry 8

Find the length of each indicated side.

Mrs. McConaughy Geometry 9

c

a

b

Pythagorean Theorem

c2 = a2 + b2

In summary: We can find the lengths of sides in right

triangles by using:30-60-90 ∆

30

2l l√3

60

l

45-45-90 ∆

l

45

l l√2

45

Pythagorean Primitives3 • 4 • 55 • 12 • 138 • 15 • 177 • 24 • 25

…and their multiples!

Mrs. McConaughy Geometry 10

Putting it all together:Find the length of each indicated side.

8 ∙ 3

8 ∙ 4

8 ∙ __ = __ 20

20√3

5 40

Mrs. McConaughy Geometry 11

Homework Assignment:

Special Right Triangles WS (1-10 all, 12)