Pythagorean Theorem Exam: Math Quiz!

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.