Pythagorean Theorem Exam: Math Quiz!

1. Ms. Coney made the triangle that you see here on the board. She said that the hypotenuse is 13 inches and one of the legs is 5 inches. What is the length of the other leg?

12 inches

12 in.

12

The length of the other leg can be found using the Pythagorean theorem, which states that 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 other two sides. In this case, the hypotenuse is 13 inches and one leg is 5 inches. By rearranging the formula and substituting the given values, we can solve for the length of the other leg. The equation becomes 5^2 + x^2 = 13^2, where x represents the length of the other leg. Solving this equation gives us x = 12 inches.

Explanation

The length of the other leg can be found using the Pythagorean theorem, which states that 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 other two sides. In this case, the hypotenuse is 13 inches and one leg is 5 inches. By rearranging the formula and substituting the given values, we can solve for the length of the other leg. The equation becomes 5^2 + x^2 = 13^2, where x represents the length of the other leg. Solving this equation gives us x = 12 inches.

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2. A suitcase measures 24 inches long and 18 inches high. What is the diagonal length of the suitcase?

30 inches

30 in.

30

The diagonal length of a rectangle can be found using the Pythagorean theorem, which states that the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides. In this case, the length and height of the suitcase form the two sides of a right triangle. By applying the Pythagorean theorem, we can calculate the diagonal length as follows: diagonal^2 = length^2 + height^2. Plugging in the given values, we get diagonal^2 = 24^2 + 18^2. Simplifying this equation gives diagonal^2 = 576 + 324, which equals 900. Taking the square root of both sides gives us the diagonal length, which is 30 inches.

Explanation

The diagonal length of a rectangle can be found using the Pythagorean theorem, which states that the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides. In this case, the length and height of the suitcase form the two sides of a right triangle. By applying the Pythagorean theorem, we can calculate the diagonal length as follows: diagonal^2 = length^2 + height^2. Plugging in the given values, we get diagonal^2 = 24^2 + 18^2. Simplifying this equation gives diagonal^2 = 576 + 324, which equals 900. Taking the square root of both sides gives us the diagonal length, which is 30 inches.

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3. Breanna has a laptop computer that has a 37-centimeter diagonal screen. The width of the laptop is 12 centimeters. What is the length of the laptop?

35 centimeters

35 cm

35

The length of the laptop can be found by using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (width and length). In this case, the width is given as 12 centimeters and the diagonal is given as 37 centimeters. By rearranging the equation and solving for the length, we can find that the length is equal to 35 centimeters.

Explanation

The length of the laptop can be found by using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (width and length). In this case, the width is given as 12 centimeters and the diagonal is given as 37 centimeters. By rearranging the equation and solving for the length, we can find that the length is equal to 35 centimeters.

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4. To get from point A to point B Myranda must avoid walking through the pond. She walks 34 meters south and 41 meters east. How many meters is the shortest distance from point A to point B?

53.3 meters

53.3 m.

53.3

53.26 meters

53.26

Myranda walks 34 meters south and 41 meters east. To find the shortest distance from point A to point B, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the southward distance and the eastward distance form the legs of a right triangle, and the shortest distance is the hypotenuse. Using the Pythagorean theorem, we can calculate the shortest distance as the square root of (34^2 + 41^2), which is approximately 53.3 meters.

Explanation

Myranda walks 34 meters south and 41 meters east. To find the shortest distance from point A to point B, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the southward distance and the eastward distance form the legs of a right triangle, and the shortest distance is the hypotenuse. Using the Pythagorean theorem, we can calculate the shortest distance as the square root of (34^2 + 41^2), which is approximately 53.3 meters.

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5. How far up a wall will a ladder reach that measures 11 meters, if the foot of the ladder must be 4 meters from the base of the wall?

10.2 meters

10.2 m.

10.2

10.24 meters

10.24

The ladder will reach a height of 10.2 meters up the wall. This can be determined using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides (the height of the wall and the distance of the foot of the ladder from the base of the wall). In this case, the ladder (11 meters) squared is equal to the height of the wall squared plus the distance from the base of the wall squared. By rearranging the equation and solving for the height of the wall, we find that it is approximately 10.2 meters.

Explanation

The ladder will reach a height of 10.2 meters up the wall. This can be determined using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides (the height of the wall and the distance of the foot of the ladder from the base of the wall). In this case, the ladder (11 meters) squared is equal to the height of the wall squared plus the distance from the base of the wall squared. By rearranging the equation and solving for the height of the wall, we find that it is approximately 10.2 meters.

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6. Two joggers run 8 miles north and then 15 miles west. What is the shortest distance, they must travel to return to their starting point?

17 miles

17 mi.

17

The shortest distance they must travel to return to their starting point can be found by using the Pythagorean theorem. The distance north and west form a right triangle, with the north distance being the vertical leg and the west distance being the horizontal leg. Using the Pythagorean theorem, we can calculate the length of the hypotenuse, which represents the shortest distance back to the starting point. In this case, the north distance is 8 miles and the west distance is 15 miles. Therefore, the shortest distance they must travel to return to their starting point is 17 miles.

Explanation

The shortest distance they must travel to return to their starting point can be found by using the Pythagorean theorem. The distance north and west form a right triangle, with the north distance being the vertical leg and the west distance being the horizontal leg. Using the Pythagorean theorem, we can calculate the length of the hypotenuse, which represents the shortest distance back to the starting point. In this case, the north distance is 8 miles and the west distance is 15 miles. Therefore, the shortest distance they must travel to return to their starting point is 17 miles.

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7. The front of a hiking tent is shaped like a triangle. The slanted sides are both 5 feet long and the bottom of the tent is 6 feet across. What is the height of the tent in feet at the tallest point?

4 feet

4 ft.

4

The height of the tent at the tallest point is 4 feet. This can be determined by using the Pythagorean theorem. The slanted sides of the triangle form a right angle with the base of the tent. By applying the Pythagorean theorem, we can find the height of the triangle. The equation is: height^2 = (slanted side)^2 - (base/2)^2. Plugging in the values, we get: height^2 = 5^2 - 3^2 = 25 - 9 = 16. Taking the square root of 16 gives us the height, which is 4 feet.

Explanation

The height of the tent at the tallest point is 4 feet. This can be determined by using the Pythagorean theorem. The slanted sides of the triangle form a right angle with the base of the tent. By applying the Pythagorean theorem, we can find the height of the triangle. The equation is: height^2 = (slanted side)^2 - (base/2)^2. Plugging in the values, we get: height^2 = 5^2 - 3^2 = 25 - 9 = 16. Taking the square root of 16 gives us the height, which is 4 feet.

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8. 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 2nd base?

127.3 feet

127.3 ft.

127.3

127.28 feet

127.28 ft.

The shortest distance between home plate and 2nd base on a baseball diamond is the diagonal of the square. Using the Pythagorean theorem, we can calculate this distance. The length of each side of the square is 90 feet, so the diagonal can be found using the formula √(90^2 + 90^2). Simplifying this equation gives us √(8100 + 8100), which equals √(16200). Rounding this value to the nearest tenth of a foot gives us 127.3 feet. Therefore, the correct answer is 127.3 feet.

Explanation

The shortest distance between home plate and 2nd base on a baseball diamond is the diagonal of the square. Using the Pythagorean theorem, we can calculate this distance. The length of each side of the square is 90 feet, so the diagonal can be found using the formula √(90^2 + 90^2). Simplifying this equation gives us √(8100 + 8100), which equals √(16200). Rounding this value to the nearest tenth of a foot gives us 127.3 feet. Therefore, the correct answer is 127.3 feet.

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9. Shaylee wanted to know if a triangle whose legs are 15 feet and 8 feet and whose hypotenuse is 17 feet is a right triangle. She put into her calculator 17x17 =289. Then she put into her calculator 15x15 =225 and 8x8 =64 and added 225 and 64 together. Is the triangle a right triangle?

No, because 15+8 does not equal 17.

No, because there is not enough information to solve this problem.

Yes, because 8x15x17 is equal to 2040.

The correct answer is "Yes, because the square of the hypotenuse is equal to the sum of the squares of the legs of the triangle." This is the correct answer because according to 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 other two sides. In this case, 17^2 is equal to 15^2 + 8^2, so the triangle is a right triangle.

Explanation

The correct answer is "Yes, because the square of the hypotenuse is equal to the sum of the squares of the legs of the triangle." This is the correct answer because according to 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 other two sides. In this case, 17^2 is equal to 15^2 + 8^2, so the triangle is a right triangle.

10. In trigonometry, the Pythagorean theorem is considered a special case of a more general law. In other words, this mathematical law is a more general formula that can be used for all types of triangles. What is the name of this law?

Law of sines

Law of tangents

Law of sines

Law of cosines

The Pythagorean theorem is a special case of the Law of Cosines. The Law of Cosines is a more general formula that can be used to find the lengths of the sides of any triangle, not just right triangles. It relates the lengths of the sides of a triangle to the cosine of one of its angles. Therefore, the Law of Cosines is the correct answer in this case.

Explanation

The Pythagorean theorem is a special case of the Law of Cosines. The Law of Cosines is a more general formula that can be used to find the lengths of the sides of any triangle, not just right triangles. It relates the lengths of the sides of a triangle to the cosine of one of its angles. Therefore, the Law of Cosines is the correct answer in this case.