1. Section 8-2 The Pythagorean Theorem Objectives: Solve problems using the Pythagorean Theorem Right Angle: angle that forms 90° Hypotenuse: in a right triangle, it is the side opposite the 90° angle Leg: each of the sides forming the right triangle Pythagorean Theorem: describes the relationship of the lengths of sides of a right triangle.

2. In 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... a 2 + b 2 = c 2 a (leg) b (leg) c (hypotenuse) 90° This is a right triangle This is NOT a right triangle The Pythagorean Theorem a and b are the legs of the right triangle c is the hypotenuse and is ALWAYS the longest segment

3. A Pythagorean Triple A set of nonzero whole numbers a, b, and c that satisfy the Pythagorean Theorem formula. If you multiply each number in a Pythagorean Triple by a whole number, the resulting numbers will also form a Triple. Common Pythagorean Triples 3, 4, 55, 12, 138, 15, 17

4. What is the length of the hypotenuse of this triangle? The Pythagorean Theorem Step 1. Write the formula for the Pythagorean Theorem. Step 2. Substitute in known values. Step 3. Solve for the unknown variable.

5. A television screen measures approximately 15.5 in. high and 19.5 in. wide. A television is advertised by giving the approximate length of the diagonal of its screen. How should this television be advertised? Real-world Pythagorean Theorem Draw and label a diagram to model the problem. 15.5 19.5 c Solve for c, using the Pythagorean Theorem. The television should be advertised as a 25”.

6. A toy fire truck is near a toy building on a table such that the Base of the ladder is 13 cm from the building. The ladder is extended 28 cm to the building. How high above the table is the top of the ladder? Real-world Pythagorean Theorem Draw a diagram to model the problem. Solve using the Pythagorean Theorem The truck is approximately (24.8 + 9) 33.8 cm above the table.

7. Determine whether the given lengths are sides of a right triangle. a. 5 in., 5 in., and 7 in. b. 10 cm, 24 cm, and 26 cm If a triangle has sides of length a and b, and a 2 + b 2 = c 2, then the triangle is a right triangle with hypotenuse of length c. Remember, c is the longest side in a right triangle. Converse of the Pythagorean Theorem Not a right triangle This is a right triangle

8. If two forces pull at right angles to each other, the resultant force is represented as the diagonal of a rectangle, as shown in the diagram. The diagonal forms a right triangle with two of the perpendicular sides of the rectangle. For a 50–lb force and a 120–lb force, the resultant force is 130 lb. Are the forces pulling at right angles to each other? Physics and the Pythagorean Theorem Yes, the forces are pulling at right angles to each other.

9. c = 16,200 Take the square root. c 127.27922 Use a calculator. a 2 + b 2 = c 2 Use the Pythagorean Theorem. 90 2 + 90 2 = c 2 Substitute 90 for a and for b. 8100 + 8100 = c 2 Simplify. 16,200 = c 2 The distance to home plate from second base is about 127 ft. Use the information to draw a baseball diamond. A baseball diamond is a square with 90-ft sides. Home plate and second base are at opposite vertices of the square. About how far is home plate from second base? Real-world Connection and the Pythagorean Theorem