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 #1

• Find the missing side of the right triangle.

Example #2

• Find the missing side of the right triangle.

Example #3

• Find the missing side of the right triangle.

Example #4

• Find the area of the right triangle.

Example #5

• Find the area of the right triangle.

53cm

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 #7

• Find the remaining two sides of each figure.

Example #8

• Find the remaining two sides of each figure.

2

3

Example #9

• Find the remaining two sides of each figure.

Example #10

• Find the remaining two sides of each figure.

3

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.

• Practice– WB 7-2 # 1, 3, 5, 10, 14-19 – WB 7-3 # 2, 4, 7, 10, 13, 15

• EXIT