Warm UP

Essential Question

How can I solve for side lengths of a right triangle and how can you use side lengths to determine whether a triangle is acute, right, or obtuse?

Lesson 41 Unit 2: Right Triangles

Section 1: The Pythagorean Theorem

Standard(s):

MCC9-12.G.SRT.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Pythagorean theorem

- The theorem states that in a right triangle , the sum of the squares of the lengths of the legs equals the square of the hypotenuse.

NOTE:

• The hypotenuse is always directly across from the 90 degree angle

We can use the Pythagorean theorem to solve for a missing side length in right triangles

Find the missing variable:1.A = 6, B= 9, C = ?2.A = 4, B = ? C = 113.A = ? B = 4, C = 134.A = 2, B = 13, C = ?5.A =7, B = ? C = 14

A

B

C

Application ProblemTo prevent a ladder from shifting, safety experts

recommend that the ratio of h:d be 4:1. How far from the base of the wall should you place the foot of a 10- foot ladder? Round to the nearest inch.

Solution

Based on the picture and what we know aboutThe Pythagorean theorem: D2 + h2 = l 2

We also know l = 10 because the Ladder is 10 feet tall

Let d have length x. Based on the ratio, h = 4x

So, (x)2 + (4x)2 = (10)2

Pythagorean triple

• A set of 3 nonzero whole numbers a, b, and c such that a2 + b2 = c2.

- The largest side length is the hypotenuse

Some common Pythagorean triples are: 3,4,55,12,138,15,177,24,25

Find the missing side length and Identify if the following is a Pythagorean triple

15

12

Find the missing side length and Identify if the following is a Pythagorean triple

15

9

In order to have a triangle

The two smaller side lengths added together must be larger than the largest side length

If they aren’t the sides can’t touch

17

1

7

17

15

8

Different kinds of trianglesRight triangles

angle C = 90 degrees c2 = a2 + b2 Obtuse triangles

Angle C > 90 degreesc2 > a2 + b2

Acute triangles angle C < 90 degrees

c2 < a2 + b2 Note: c is the longest side of the triangle

Tell if the measures can be the side lengths of a triangle. If so, Classify the triangle as acute, obtuse, or right

a. 8,11,13

b. 5.8, 9.3, 15.6