7B Pythagorean Theorem and Its Converse

OBJECTIVES:To determine missing measures using the Pythagorean TheoremTo determine right triangles using the Converse of the Pythagorean Theorem

Right Triangle Parts

Longest sideOpposite rt. angle

THEOREMTHEOREM: Pythagorean TheoremIn a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Right ∆ c2 = a2 + b2

NOTE: The Pythagorean Theorem is useful in finding missing lengths of sides in right triangles

hypotenuseleg

leg

Using the Pythagorean Theorem

EXAMPLE 1: Finding the Length of a Hypotenuse

Given a right triangle with legs of lengths 5 cm and 12 cm, find the length of the hypotenuse.

Using the Pythagorean Theorem

EXAMPLE 2: Finding the Length of a Leg

Given a right triangle with hypotenuse of length 14 cm and leg of length 7 cm, find the length of the remaining leg.

Using the Pythagorean Theorem

Find the area of the triangle at the left to the nearest tenth of a square meter.

Recall: In an isosceles triangle, the height is the median is the angle bisector.

THEOREMTHEOREM: Converse of the Pythagorean TheoremIf the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

c2 = a2 + b2 right triangle

NOTE: The Converse of the Pythagorean

Theorem is useful in determining

right triangles.

The Pythagorean Theorem and Its Converse can be written as the following bi-conditional

statement:

Right ∆ c2 = a2 + b2

8 7 4√95 15

√113 36

The triangles below appear to be right triangles. Determine whether they are right triangles.

Using the Converse of the Pythagorean Theorem:

EXAMPLE 4: Determining Right Triangles

THEOREMSTHEOREMS TO DETERMINE ACUTE OR OBTUSE

TRIANGLESIf the square of the length of the longest side of a

triangle is ____________ the sum of the squares of the lengths of the other two sides, then the triangle is an

__________ triangle. c2 < a2 + b2 acute triangle

If the square of the length of the longest side of a triangle is ____________ the sum of the squares of the lengths of the other two sides, then the triangle is an

__________ triangle. c2 > a2 + b2 obtuse triangle

EXAMPLE 5: Classifying Triangles

1. Determine if a triangle can be formed given the following lengths of sides.

2. If they can, classify the triangle as right, acute, or obtuse.a. 38 cm, 77cm, 86cm b. 10.5cm, 36.5cm,

37.5cm

To summarize:

Pythagorean Theorem and Its Converse

Right ∆ __________________

c2 = a2 + b2 ____________Classifying Right Trianglesc2 < a2 + b2 ____________c2 > a2 + b2 ____________

Final Checks for Understanding

1. State the Pythagorean Theorem in your own words.

2. Which equations are true for ∆ PQR?

a. r2= p2 + q2

b. q2= p2 + r2

c. p2= r2 - q2

d. r2= (p + q)2

e. p2= q2 + r2

Q

r p

P q R

Final Checks for Understanding

3. State the Converse of the Pythagorean Theorem in your own words.

4. Match the lengths of the sides with the appropriate description.

5. 2, 10, 116. 13, 5, 77. 5, 11, 68. 6, 8, 10

A. right ∆B. acute ∆C. obtuse right ∆D. not a ∆

HOMEWORK ASSIGNMENT:

Pythagorean Theorem and Its Converse WS, plus textbook:_______________________