Geometry
• Agenda1. ENTRANCE2. Go over Tests/Spiral3. 7-2 The Pythagorean Theorem and its Converse4. 7-3 Special Right Triangles5. Practice Assignment6. EXIT
Chapter 9
(We actually start with 2 sections of Chapter 7.)
7-2 The Pythagorean Theorem and its Converse
7-3 Special Right Triangles
Theorem 7-4The Pythagorean Theorem
• In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
2 2 2a b c
Common Pythagorean Triples
• Certain sets of three numbers appear often in Geometry problems since they satisfy the Pythagorean Theorem.
– 3, 4, 5– 5, 12, 13 Multiples of these triples– 8, 15, 17 will work as well, such as– 7, 24, 25 6, 8, 10 and 15, 36, 39.– 9, 40, 41
Theorems 7-5, 7-6, and 7-7Converse of the Pythagorean Theorem• If , then the triangle is a right
triangle.
• If , then the triangle is an obtuse triangle.
• If , then the triangle is an acute triangle.
2 2 2c a b
2 2 2c a b
2 2 2c a b
Example #6
• What type of triangle are each of the following?– A. 4, 6, 7 E. 8, 8, 8
– B. 15, 20, 25 F. 16, 48, 50
– C. 10, 15, 20 G. 7, 8, 9
– D. 13, 84, 85 H. 6, 11, 14
Theorem 7-845°-45°-90° Triangle Theorem
• In a 45°-45°-90° triangle, both legs are congruent and the length of the hypotenuse is
times the length of a leg. 2
45° 45° 90°n n n 2
2
Theorem 7-930°-60°-90° Triangle Theorem
• In a 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is times the length of the shorter leg.
3
30° 60° 90°n n 2n3
3
Example #11
• A square garden has sides 100 ft long. You want to build a brick path along a diagonal of the square. How long will the path be?
Example #12
• The distance from one corner to the opposite corner of a square playground is 96 ft. How long is each side of the playground?
Example #13
• A garden shaped like a rhombus has a perimeter of 100 ft and a 60° angle. Find the area of the garden.
Example #14
• A rhombus has 10-inch sides, two of which meet to form a 30° angle. Find the area of the rhombus.