Pythagorean Theorem MCQ Exam!

Explanation
The distance between point A and point B would be 53. This is because the question states that in order to get from point A to point B, one must avoid walking through a pond. Therefore, the distance does not include the distance across the pond.

Explanation
The shortest distance between first base and third base on a baseball diamond can be calculated using the Pythagorean theorem, since the bases form a right triangle. The distance between first base and third base is the hypotenuse of the right triangle, and the two sides are the distances between first base and second base (90 feet) and between second base and third base (90 feet). Therefore, the shortest distance is the square root of (90^2 + 90^2), which is approximately 127.3 feet.

Explanation
The diagonal length of a rectangular box can be found using the Pythagorean theorem, which states that the square of the length of the diagonal is equal to the sum of the squares of the other two sides. In this case, the length is 24 inches and the height is 18 inches. Using the theorem, we can calculate the square of the diagonal length as 24^2 + 18^2 = 576 + 324 = 900. Taking the square root of 900 gives us the diagonal length, which is 30 inches. Since the question asks for the nearest tenth of a foot, we convert 30 inches to feet by dividing by 12, giving us 2.5 feet. Therefore, the correct answer is 30.0.

Explanation
The pattern in the given sequence is that each number is obtained by adding 2 to the previous number. Starting with 6, we add 2 to get 8, and this pattern continues with 10 and 12. Therefore, the missing number in the sequence is 8.

Explanation
Tanya runs diagonally across the rectangular field, which means she runs from one corner to the opposite corner. The length of the diagonal can be found using the Pythagorean theorem, which 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 length of one side is 40 yards and the width is 30 yards. Therefore, the length of the diagonal can be found by taking the square root of (40^2 + 30^2), which is equal to 50 yards.

Explanation
After walking 6 blocks North and then 8 blocks West, John has formed a right-angled triangle. The distance from the school to John's current position can be calculated using the Pythagorean theorem. The length of the hypotenuse (distance from the school) is equal to the square root of the sum of the squares of the other two sides. In this case, the hypotenuse is equal to the square root of (6^2 + 8^2), which simplifies to the square root of (36 + 64), resulting in a distance of 10 blocks from the school.

Explanation
The ladder is placed 5 feet away from the wall, and the distance from the ground straight up to the top of the wall is 13 feet. This forms a right triangle with the ladder as the hypotenuse. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the square of the hypotenuse (13^2) is equal to the sum of the squares of the other two sides (5^2 + x^2), where x is the height of the ladder on the wall. Solving this equation, we find that x^2 = 144, and taking the square root of both sides, we get x = 12. Therefore, the ladder reaches the top of the wall.

Explanation
The length of the wire can be found using the Pythagorean theorem, which states that in a right-angled 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 pole acts as the hypotenuse, the distance from the base of the pole to the wire is one side, and the length of the wire is the other side. Using the theorem, we can calculate the length of the wire as √(16^2 - 12^2) = √(256 - 144) = √112 = 10.6, which is approximately equal to 20.

Explanation
If the distance between Bob and the top of the flagpole is 15 ft and the flagpole is 12 ft tall, then Bob must be standing 3 ft away from the base of the flagpole (15 ft - 12 ft = 3 ft). Therefore, Bob is 9 ft away from the base of the flagpole (12 ft - 3 ft = 9 ft).

Explanation
The distance between the base of the ladder from the building can be found using the Pythagorean theorem. The ladder, the wall, and the ground form a right triangle. The length of the ladder is the hypotenuse of the triangle, which is 12 meters. The height of the triangle is 10 meters. To find the distance between the base of the ladder and the building, we need to find the length of the base of the triangle. Using the Pythagorean theorem, we can calculate the length of the base to be approximately 6.6 meters.