1. 7B Pythagorean Theorem and Its Converse
OBJECTIVES:
To determine missing measures using the Pythagorean Theorem
To determine right triangles using the Converse of the Pythagorean Theorem

2. Right Triangle Parts
Longest side
Opposite rt. angle

3. THEOREM THEOREM: Pythagorean Theorem
NOTE: The Pythagorean Theorem is useful in finding missing lengths of sides in right triangles
THEOREM
THEOREM: Pythagorean Theorem
In 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
hypotenuse
leg
leg

4. 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.

5. 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.

6. 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.

7. THEOREM THEOREM: Converse of the Pythagorean Theorem
NOTE: The Converse of the
Pythagorean
Theorem
is useful in determining
right triangles.
THEOREM
THEOREM: Converse of the Pythagorean Theorem
If 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

8. The Pythagorean Theorem and Its Converse can be written as the following bi-conditional statement:
Right ∆ c2 = a2 + b2

9. Using the Converse of the Pythagorean Theorem:
EXAMPLE 4: Determining Right Triangles
The triangles below appear to be right triangles. Determine whether they are right triangles. √ √

10. THEOREMS c2 < a2 + b2  acute triangle
THEOREMS TO DETERMINE ACUTE OR OBTUSE TRIANGLES
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  acute triangle
c2 > a2 + b2  obtuse triangle

11. EXAMPLE 5: Classifying Triangles
Determine if a triangle can be formed given the following lengths of sides.
If they can, classify the triangle as right, acute, or obtuse.
a. 38 cm, 77cm, 86cm b. 10.5cm, 36.5cm, 37.5cm

12. To summarize: Pythagorean Theorem and Its Converse
Right ∆  __________________
c2 = a2 + b2  ____________
Classifying Right Triangles
c2 < a2 + b2  ____________
c2 > a2 + b2  ____________

13. Final Checks for Understanding
State the Pythagorean Theorem in your own words.
Which equations are true for ∆ PQR?
a. r2= p2 + q2
b. q2= p2 + r2
c. p2= r2 - q2
d. r2= (p + q)2 Q
r p
P q R
e. p2= q2 + r2

14. 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, 11
6. 13, 5, 7
7. 5, 11, 6
8. 6, 8, 10
A. right ∆
B. acute ∆
C. obtuse right ∆
D. not a ∆

15. HOMEWORK ASSIGNMENT:
Pythagorean Theorem and Its Converse WS, plus textbook:_______________________