Working with  Pythagorean Theorem Topic Index | Geometry Index | Regents Exam Prep Center
 Answer the following questions dealing with the Pythagorean Theorem.
 1. Princess Fiona is locked in the tower of Castle Kronen.  You have volunteered to rescue the princess.  If the tower window is 36 feet above the ground and you must place your ladder 10 feet from the  base of the castle (because of the moat), which choice is the shortest length ladder you will need to reach the tower window? Choose: 34 feet 35 feet 37 feet 38 feet Explanation Set up your triangle as indicated in the picture. The missing side is c, the hypotenuse. Utilize the theorem c²=a²+b². The length of the hypotenuse is 37.36308338 ft. The choice must be the 38 feet ladder. All other choices are too short to cover the distance.

 2. A baseball diamond is a square with sides of 90 feet.  What is the shortest distance, to the nearest tenth of a foot, between home plate and second base? Choose: 90.0 127.3 180.0 180.7 Explanation Use the Pythagorean Theorem. a and b both equal 90. Solving for c gives c = 127.3.

 3. The sides of a triangle measure 2.4, 3.2, and 4.  Is this triangle a right triangle? Choose: YES NO Explanation If the sides satisfy the Pythagorean Theorem, the triangle will be a right triangle. 4² = 2.4² + 3.2² 16 = 16 Yes it is a right triangle.

 5. Using the Pythagorean Theorem, find the area of an equilateral triangle whose side measures 5 units.  Find the area to the nearest tenth of a square unit. Choose: 4.3 6.5 10.8 12.5 Explanation Find the height first. 5² = 2.5² + h² h² = 18.75 Find the area: A = (1/2)bh Area = (1/2)(5)(square root of 18.75)

 6. An equilateral triangle is plotted on a coordinate plane.  Two of the vertices are (0,0) and (8,0).  Which of the coordinates shown could be the vertex of the third side? Choose: Explanation Draw a graph to get an understanding of the problem. The third vertice will lie on the line x = 4. By drawing the altitude, a right triangle is formed whose hypotenuse is 8 and whose known leg is 4. Use the theorem to find the altitude and get the y-value. The answer could actually be either positive or negative for the y-value, but only the positive value is shown.

 7. Joe Bean regularly takes a short-cut across Mr. Wilson's lawn instead of walking on the sidewalk on his way home from school.  How much distance is saved by Joe cutting across the lawn? Choose: 5 ft 10 ft 25 ft 35 ft Explanation The triangle is a Pythagorean Triple. 3, 4, 5 multiplied by 5. The shortcut path is 25 feet. If Joe walked on the sidewalk, the distance is 35 feet. He saves 10 feet by taking the shortcut.

 8. Find x. Choose: Explanation Continue around the figure. Notice the pattern that starts to develop. Find the hypotenuse of the first triangle. Use this value to find the next hypotenuse.

 9. Find the value of x. Choose: 3 4 5 6 Explanation Using the Pythagorean Theorem: 20² = (3x)² + (4x)² Watch out for those parentheses! 400 = 9x² + 16x² 400 = 25x² 16 = x² 4 = x OR ................ Use your knowledge of Pythagorean Triples to solve this problem. The hypotenuse of 20 will = 5x.

 10. A spider has taken up residence in a small cardboard box which measures 2 inches by 4 inches by 4 inches.  What is the length, in inches,  of a straight spider web that will carry the spider from the lower right front corner of the box to the upper left back corner of the box? Choose: Explanation Start by finding the diagonal of the bottom of the box. Use this diagonal distance to form a diagonal right triangle.

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