1. The Right Triangle and The Pythagorean Theorem

2. Verify the Pythagorean Theorem; and Apply the Pythagorean Theorem
Objective
Standard 8.10
The student will
Verify the Pythagorean Theorem; and
Apply the Pythagorean Theorem

3. What is a Triangle?
A triangle is any geometric shape consisting of three points or vertices which are connected by straight line segments called sides.

4. What is a right triangle?
leg
hypotenuse
right angle
leg
It is a triangle which has an angle that is 90 degrees.
The two shortest sides that make up the right angle are called legs.
The longest side opposite the right angle is the hypotenuse.

5. Pythagorean Theorem
Over 2,500 years ago, a Greek mathematician named Pythagoras popularized the concept that a relationship exists between the hypotenuse and the legs of right triangles and that this relationship is true for all right triangles. He said “In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs”. This is known as the Pythagorean Theorem.

6. The Pythagorean Theorem
In a right triangle, if a and b are the measures of the legs and c is the hypotenuse, then a2 + b2 = c2. Note: The hypotenuse, c, is always the longest side.

7. You can use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle when you are given the lengths of the two legs.

8. Finding the Hypotenuse
Find the length of the hypotenuse of a right triangle whose legs are 5 cm and 12 cm. a2 + b2 = c = c = c2 169 = c2 13 = c The length of the hypotenuse is 13 cm.
Write the Pythagorean formula. Substitute in the values for the legs a, and b. Add the legs together. Simplify. Take the square root of each side to remove the c2 and solve for c.

9. Finding the Hypotenuse
Find the length of the hypotenuse of a right triangle whose legs are 12 cm and 16 cm. a2 + b2 = c = c = c2 400 = c2 20 = c The length of the hypotenuse is 20 cm.
Write the Pythagorean formula. Substitute in the values for the legs a, and b. Add the legs together. Simplify. Take the square root of each side to remove the c2 and solve for c.

10. Find the length of the hypotenuse if a = 5 and b = 7.
= c2
= c2
74 = c2
Take the square root of both sides.
8.60 = c

11. Find the length of the hypotenuse given a = 6 and b = 12
180
324
13.42 18

12. Let’s Try A Few On Your Own…
Find the length of the hypotenuse if the triangle has leg lengths of 6 inches and 3 inches.
Find the length of the hypotenuse given a = 4, and b = 10.
Find the length of the hypotenuse given a = 3, and b = 8.
Find the length of the missing side:
Label the parts of the right triangle for the diagram.

13. You can use the Pythagorean Theorem to find the length of a leg as well.

14. Finding a Leg of a Right Triangle
The hypotenuse of a right triangle is 15 cm and one leg is 9 cm. Find the length of the other leg.
a2 + b2 = c2
92 + b2 = 152
81 + b2 = 225
b2 = 144
b = 12
length is 12 cm
Write the Pythagorean formula. Substitute in the values for the leg a, and the hypotenuse c. Get b2 on the side by itself. Simplify. Take the square root of each side to remove the b2 and solve for b.

15. Find the length of the leg, to the nearest hundredth, if a = 4 and c = 10.
42 + b2 = 102
16 + b2 = 100
Solve for b.
b2 =
b2 = 84
b = 9.17

16. Let’s Try A Few On Your Own…
Find the length of the leg to the nearest hundredth, if one leg is 3 cm and the hypotenuse is 5 cm.
Find the length of the leg to the nearest hundredth, if one leg is 10 inches and the hypotenuse is 12 inches.
Find the length of the leg to the nearest hundredth, if one leg is 12 inches and the hypotenuse is 15 inches.
Find the length of the missing side:

17. If the equation a2 + b2 = c2 is true for the lengths of the sides of a triangle, then the triangle is a right triangle. This method is called the converse of the Pythagorean theorem.

18. Is a triangle with sides 7 in., 25 in., and 24 in., a right triangle?
If the triangle fits the Pythagorean theorem it is a right triangle. The longest side is the hypotenuse. a2 + b2 = c = = 625 Is the equation true? Yes, so it is a right triangle

19. Is a triangle with sides 6 cm., 8 cm., and 10 cm., a right triangle?
If the triangle fits the Pythagorean theorem it is a right triangle. The longest side is the hypotenuse. a2 + b2 = c = = 100 Is the equation true? Yes, so it is a right triangle

20. The measures of three sides of a triangle are given below
The measures of three sides of a triangle are given below. Determine whether this triangle is a right triangle , 3, and 8
Which side is the biggest?
The square root of 73 (= 8.5)! This must be the hypotenuse (c).
Plug your information into the Pythagorean Theorem. It doesn’t matter which number is a or b.

21. Sides: , 3, and = ( ) 2
= 73
73 = 73
Since this is true, the triangle is a right triangle!! If it was not true, it would not be a right triangle.

22. Determine whether the triangle is a right triangle given the sides 6, 9, and
Yes No
Purple

23. Let’s Try A Few On Your Own…
Determine whether the triangle given the sides 8, 14, and 13 is a right triangle.
Determine whether the triangle given the sides square root of 25, square root of 36, and square root of 81 is a right triangle.
Determine whether the triangle is a right triangle?