5.4 The Pythagorean Theorem & Converse Es sential Question: How can we identify and apply the Pythagorean Theorem and its converse by solving problems.

5.4 The Pythagorean Theorem & Converse

Essential Question: How can we identify and apply the Pythagorean Theorem and its converse by solving problems that have real world applications?

Vocabulary

Pythagorean Theorem

Converse of the Pythagorean Theorem

Pythagorean Inequalities

BELLRINGER

Find the midpoint of the segment with endpoints ( 9 , 8 ) and ( 3 , 5 ).

*REMEMBER: THE MIDPOINT FORMULA*

Pythagorean Theorem

For any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

a2 + b2 = c2

a

c

b

Pythagorean Theorem (Guided Practice)

The legs of a right triangle are 2 cm and 6 cm. Find the hypotenuse.

The legs of a right triangle are 7 cm and 24 cm. Find the hypotenuse.

One leg of a right triangle is 3 cm, the hypotenuse is 5cm. Find the missing leg.

When climbing a ladder, the rule of thumb to follow regarding safety is:

The distance from the bottom of the ladder should be 1/4 the length of the ladder

1.) a.) If you have a 16 foot ladder how far away from the wall should the bottom of the ladder be so you are safe?

b.) How far up will the ladder reach?

Converse of the Pythagorean Theorem

If the square of the length of one side of a triangle equals the sum of the squares of the lengths of the two other sides, then the triangle is a right triangle.

1.2 miles

3.5

mile

s

C

A farmer decides to plow his field into the shape of a triangle. If the sides have the lengths shown in the figure, is the field being cut into a right triangle? Use the Pythagorean Theorem to justify your answer.

Pythagorean Inequalities

For ABC, with c as the length of the longest side:

If c2 = a2 + b2 then ABC is a right triangle.If c2 a2 + b2 then ABC is an obtuse triangleIf c2 a2 + b2 then ABC is an acute triangle

Pythagorean Inequalities (Guided Practice)

1.) 122 _____ 72 + 82

2.) 142 _____ 72 + 92

3.) 22 _____ 32 + 3.52

A baseball diamond is a square with 90-foot sides.

What is the approximate distance of a catcher's throw from home plate to second base?

What is the approximate distance of the third baseman's throw to first base?

EXIT TICKET

1.) A right triangle has one leg with a length of 48 cm and a hypotenuse (c) with a length of 80. What is the length of the other leg?

2.) A triangle has leg lengths of 8 cm and 15 cm, and a hypotenuse of 18 cm. Is the triangle a right, obtuse, or acute triangle?

48 cm

80 cm