1. Shown below is a Geometer’s Sketchpad screen 1. Shown below is a Geometer’s Sketchpad screen. In the diagram, ABC is an isosceles right triangle with right angle B. The points plotted are data from the table (AB, AC). c. The slope of the line you graphed in question b is not precise (it has been rounded off by Sketchpad). What is the exact value of the slope? d. Using your prior knowledge of right triangles, explain how you could have found the slope without graphing any points.

b. Conjecture a relationship between the length of altitude and the product (AP)(PB). c. Prove your conjecture using similar triangles and proportions. Shown below is a Geometer’s Sketchpad screen. In the diagram, ABC is a right triangle and is an altitude on hypotenuse . When the altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean between the segments of the hypotenuse. Cross multiplication – The product of the means equals the product of the extremes. PC2 = (AP)(PB) extremes means

b. Conjecture a relationship between the length of altitude and the product (AP)(PB). c. Prove your conjecture using similar triangles and proportions. Shown below is a Geometer’s Sketchpad screen. In the diagram, ABC is a right triangle and is an altitude on hypotenuse . When the altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean between the segments of the hypotenuse. Cross multiplication – The product of the means equals the product of the extremes. PC2 = (AP)(PB)

When the altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean between the segments of the hypotenuse APC  BPC (right angles) Because ACB is a right angle, ACP is complementary to BCP PAC is complementary to ACP (they are the acute angles of a right triangle) PAC  BCP because they are complementary to the same angle. APC~ CPB (AA~) Definition of similar triangles PC2 = (AP)(PB) 

Infinitely many rectangles with different dimensions have an area of 36 square units (e.g. 3x12, 4x9, 6x6, 8x4½, 10x3.6, 10x1.6, 15x2.4 to name a few). Use Geometer’s Sketchpad to construct a rectangle whose area is 36, and which retains that area when the dimensions are changed by dragging its vertices. (Hint – question 2 above can help you in this construction.) When the altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean between the segments of the hypotenuse. or a b h

4. In the diagram, ABCD is a rectangle and is perpendicular to . Prove: 

We can also prove that each smaller triangle is similar to ABC. When the altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean between the segments of the hypotenuse From APC ~ CPB When the altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg From APC ~ ACB From BPC ~ BCA We can also prove that each smaller triangle is similar to ABC.

We can also prove that each smaller triangle is similar to ABC. When the altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean between the segments of the hypotenuse From APC ~ CPB When the altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg From APC ~ ACB From BPC ~ BCA We can also prove that each smaller triangle is similar to ABC.

We can also prove that each smaller triangle is similar to ABC. When the altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean between the segments of the hypotenuse From APC ~ CPB When the altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg From APC ~ ACB From BPC ~ BCA We can also prove that each smaller triangle is similar to ABC.

We can also prove that each smaller triangle is similar to ABC. When the altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean between the segments of the hypotenuse From APC ~ CPB When the altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg From APC ~ ACB From BPC ~ BCA We can also prove that each smaller triangle is similar to ABC.

We can also prove that each smaller triangle is similar to ABC. When the altitude is drawn to the hypotenuse of a right triangle, the altitude is the geometric mean between the segments of the hypotenuse From APC ~ CPB When the altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg From APC ~ ACB From BPC ~ BCA We can also prove that each smaller triangle is similar to ABC.

The Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. on the on the B C A

Proof of the Pythagorean Theorem B C A P Prove:

Proof of the Pythagorean Theorem When the altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg B C A P Prove:

Proof of the Pythagorean Theorem AB AC AB BC = = and AC AP BC PB AC2 = (AB)(AP) and BC2 = (AB)(PB) AC2 + BC2 = (AB)(AP) + (AB)(PB) B C A P AC2 + BC2 = (AB)(AP+ PB) AC2 + BC2 = (AB)(AB) AC2 + BC2 = AB2 Prove:

The Converse of the Pythagorean Theorem is also true: If the side lengths of a triangle are a, b, and c, and , …. , then the triangle is a right triangle. B C A

Pythagorean Theorem applications In questions 1 and 2, find the length of the side marked x to the nearest tenth. A B C 5 8 2. x A B C 1. x 7 24 25 6.2 15 cm 3. What is the length of the diagonal of the rectangle shown? 17 cm 8 cm 13 in 5 in 16 in BE is an altitude of ABC. Find the perimeter and area of ABC. 4. P = 54 inches A = 126 sq inches

30 60 2a a 45 a

5. One of the angles of a rhombus measures 45, and its sides are 20 cm long. What is the area of the rhombus? Answer to the nearest tenth of a square cm. 282.8 sq cm 60 6. What is the area of quadrilateral ABCD? Answer to the nearest tenth of a square inch. 10" 136.6 sq " 7. The hypotenuse of a right triangle is 1 inch longer than its longer leg. If the shorter leg is 9 inches long, what is the length of the longer leg? 40 inches