Name: Pythagorean Theorem February 3, 2014

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Julius Anthony
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1 1. John leaves school to go home. He walks 6 blocks North and then 8 blocks west. How far is John from the school? 5. A 26 foot long ladder is leaning up against a house with its base 10 feet away from the house on the ground. If the side of the house is perfectly vertical, how far up the house would you be if you climbed to the top of the ladder? 2. What is the measure of angle A, in degrees? 6. If you walk 50 yards south, then 40 yards east, and finally 20 yards north, how far are you from your starting point? Express your answer in yards. 7. A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule: a² + b² = c². The smallest Pythagorean Triple is 3, 4 and 5 because 3² + 4² = 5². Name at least 2 other Pythagorean Triples. 3. Find the area of this right triangle. 4. A rectangle 4 cm by 12 cm is divided into four triangles, as shown. The areas of three of the four triangles are stated in the diagram. Find the number of sq cm in the area of the shaded triangle. 8. A twenty-foot high spotlight is being shined towards the top of your head, creating a shadow. If you are five feet tall, and you are standing twelve feet away from the spotlight, how long is your shadow? 16 sq cm 18 sq cm 8 sq cm Cascade Ridge PTSA Math Club 1 Show your work on a separate sheet of paper.

2 BONUS PROBLEMS 9. What is the length of the hypotenuse for the longest triangle in the figure? (All triangles are right triangles) 10. Find the area of rectangle ABCD. 11. Square ACEG is drawn below. Points B, D, F, and H are the midpoints of the sides of a square. What is the total number of squares of all sizes that can be traced using only the line segments shown? A B C H D G F E 12. The following geometric design is constructed by adding new squares to each rectangle. To the nearest whole number, what would be the length of the longest diagonal, if the two smallest squares have side-lengths of three? Cascade Ridge PTSA Math Club 2 Show your work on a separate sheet of paper.

3 SOLUTIONS: blocks The distance from school to home is the length of the hypotenuse. Let c be the missing distance from school to home and a = 6 and b = 8 c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 100 c = 100 c = The distance from school to home is 10 blocks. The sum of the angles in any triangle is 180. We already have a 90 angle and a 14 angle, so the remaining angle is = cm 2 First find the length of the missing side: = 36 = a². The missing side is 6cm long, so the area is ½ * 8 * 6 or 24 cm² 4. 6 cm² The area of a rectangle is obtained by multiplying the length by the width. If the rectangle measures 4cm by 12 cm, the total area is 48 cm². The sum of the areas of the other 3 triangles is or 42 cm². The area of the shaded triangle is or 6 cm² feet The ladder, the side of the house, and the ground form three sides of a right triangle as shown: Cascade Ridge PTSA Math Club 3 Show your work on a separate sheet of paper.

Using Pythagorean Theorem, a 2 + b 2 = c 2 10 2 + b 2 = 26 2 100 + b 2 = 676 b 2 = 576 b =

you can use the Pythagorean Theorem: a 2 + b 2 = c 2 30 2 + 40 2 = c 2 900 + 1600 = c 2

4 Using Pythagorean Theorem, a 2 + b 2 = c b 2 = b 2 = 676 b 2 = 576 b = yards If you draw it out, it looks like this: A right triangle is formed, and now you can use the Pythagorean Theorem: a 2 + b 2 = c = c = c = c 2 c = 50 Cascade Ridge PTSA Math Club 4 Show your work on a separate sheet of paper.

7. Possible answers include (5,12,13), (9,12,15), (6,8,10), (10,24,26) or (15,36,39), though the possibilities are infinite. 8. 4 ft The big triangle is similar to the gray triangle.

5 7. Possible answers include (5,12,13), (9,12,15), (6,8,10), (10,24,26) or (15,36,39), though the possibilities are infinite ft The big triangle is similar to the gray triangle. So the ratio of the sides are equal: a 5 = a Taking the cross product, 5 ( a + 12 ) = 20 a 5a + 60 = 20a 60 = 15a a = 4 Drawing a chart will help reveal the pattern. Leg 1 Leg 2 Hypotenuse Cascade Ridge PTSA Math Club 5 Show your work on a separate sheet of paper.

6 1 3 4 = = Using the Pythagorean Theorem, we can find that side (DC)² = 3² + 4² = 25, so DC = 5. We also know that the area of the triangle is 1/2 base x height. In this case, the base is 5, and the height would be the same as AD or BC. The area of the rectangle would be simply base x height, or 5 x either AD or BC. However, since we don t know the measure of AD or BC, we need to find the area of the triangle a different way, knowing that it will be equal to ½ the area of the rectangle ABCD. Imagine rotating the rectangle so that the measurement of 3 becomes the base, and 4 becomes the height of the triangle. Now we can calculate the area of the triangle: ½ 3 4 = 6. Therefore, the area of the rectangle is 12. The diagram is the result of combining these two figures: Each has 4 small squares and 1 large square for a total of 5 squares = Following the geometric pattern, we can see that the smallest squares have side lengths of 3, the next has a length of 6, the next is 9, then 15, then 24, then 39. So the length of the largest square s diagonal would be = 3042 = 55. Cascade Ridge PTSA Math Club 6 Show your work on a separate sheet of paper.

Name: Pythagorean theorem February 4, 2013

Name: Pythagorean theorem February 4, 203 ) If you walk 50 yards south, then 40 yards east, and finally 20 yards north, how far are you from your starting point? Express your answer in yards. 6) At twelve