2 “The Pythagorean Theorem” (10-2) What are the parts of a right triangle? What is the Pythagorean Theorem?hypotenuse: The side opposite of the right angle (the longest side of a right triangle) legs: the sides that form the right angle (the shortest sides of the triangle) Pythagorean Theorem: In any right triangle, the sum of the squares of the length of the legs, a and b, is equal to the square of the length of the hypotenuse, c. Note: The legs terms, “a” and “b” are interchangeable (can be switched) since you’re adding them together (Commutative Property), but the hypotenuse is always the “c” term.
3 What is the length of the hypotenuse of this triangle? The Pythagorean TheoremLESSON 10-2Additional ExamplesWhat is the length of the hypotenuse of this triangle?a2 + b2 = c2 Use the Pythagorean Theorem.= c2 Substitute 8 for a and 15 for b.= c2 Simplify.289 = c2 Find the principal square root of each side.17 = c Simplify.The length of the hypotenuse is 17 m.
4 Define: Let b = height (in cm) of the ladder from The Pythagorean TheoremLESSON 10-2Additional ExamplesA toy fire truck is positioned so that the base of the ladder is 13 cm from the wall. The ladder is extended 28 cm to the wall. How high above the table is the top of the ladder?Define: Let b = height (in cm) of the ladder froma point 9 cm above the table.Words: The triangle formed is a right triangle.Use the Pythagorean Theorem.
5 b2 = 615 Subtract 169 from each side. The Pythagorean TheoremLESSON 10-2Additional Examples(continued)Translate: a2 + b2 = c2132 + b2 = 282 Substitute.169 + b2 = 784 Simplify.b2 = 615 Subtract 169 from each side.b2 = Find the principal square root of each side.b Use a calculator and round to thenearest tenth.The height to the top of the ladder is 9 cm higher than 24.8 cm,so it is about 33.8 cm from the table.
6 “The Pythagorean Theorem” (10-2) What is the “Converse (Opposite) of the Pythagorean Theorem”?Converse of the Pythagorean Theorem: If a triangle has sides a, b, and c, and a2 + b2 = c2 (in other words, if the sum of the squares of the shorter sides equal the square of the longest side), then the triangle is a right triangle with hypotenuse c. The Converse (Opposite) of the Pythagorean Theorem can be use to determine whether or not a triangle is a right triangle. If the Pythagorean Theorem is a true statement after substituting the lengths of the sides, the triangle is a right triangle Example: Determine whether the given lengths can form a right triangle. 5 in., 12 in., and 13 in. 7 in., 9 in., and 12 in.Make a and b the two shorter sides.Use the Converse of the Pythagorean Theorem.The triangle is a right triangle.The triangle is not a right triangle.
7 Determine whether the given lengths can be sides of a right triangle. The Pythagorean TheoremLESSON 10-2Additional ExamplesDetermine whether the given lengths can be sides of a right triangle.a. 5 in., 5 in., and 7 in.Determine whether a2 + b2 = c2,where c is the longest side.Simplify.50 = 49/This triangle is not a right triangle.b. 10 cm, 24 cm, and 26 cmDetermine whether a2 + b2 = c2,where c is the longest side.Simplify.676 = 676This triangle is a right triangle.
8 If two forces pull at right angles to each other, the The Pythagorean TheoremLESSON 10-2Additional ExamplesIf two forces pull at right angles to each other, theresultant force is represented as the diagonal of a rectangle, asshown in the diagram. The diagonal forms a right triangle with twoof the perpendicular sides of the rectangle.For a 50–lb force and a 120–lb force, the resultant force is130 lb. Are the forces pulling at right angles to each other?Determine whether a2 + b2 = c2where c is the greatest force., ,90016,900 = 16,900The forces of 50 lb and 120 lb are pulling at right angles to each other.
9 1. Find the missing length 2. Find the missing length The Pythagorean TheoremLESSON 10-2Lesson Quiz1. Find the missing length 2. Find the missing lengthto the nearest tenth. to the nearest tenth.3. A triangle has sides of lengths12 in., 14 in., and 16 in. Is thetriangle a right triangle?4. A triangular flag is attached to a post. The bottom of the flag is 48 in.above the ground. How far from the ground is the top of the flag?16.65.7no57 in.