Lesson 6, Unit 4:

Converse of Pythagorean Theorem Practice Answers

1. Which set of numbers represents a Pythagorean triple?
Answer:
D. 27,36,45

2. Of the lengths of the legs in I right triangle are 7 and 8, what is the approximate length of the hypotenuse?
Answer:
D. 10.6

3. Matt is repairing a gate and needs to nail a brace diagonally to strengthen the posts. If the height of the post is 48 inches and there is 64 inches between posts, in inches, what should the length of the diagonal brace be in order to fit between the posts?
Answer:
A. 80

4. Bashir works in a weather station has launched a weather balloon. The height of the balloon is 4 kilometers and it is 1 kilometers from the station. In kilometers, what is the approximate distance between the station and the balloon?
Answer:
C. 4.1

thank chu

1. Ah, yes, a Pythagorean triple! Just like the triplets you see on the news, but with numbers instead of babies. In this case, we have 27, 36, and 45. So the answer is D, because apparently these numbers get along better than most people do.

2. Alright, let's put those legs to work and find the hypotenuse! Using the Pythagorean theorem, we have 7 squared plus 8 squared equals the hypotenuse squared. After a bit of math magic, we find that the approximate length of the hypotenuse is 10.6. So, the answer is D, because apparently hypotenuses like to keep people on their toes.

3. Ah, the magical world of gate repairing. Now, we have a height of 48 inches and a distance between the posts of 64 inches. We're looking for the length of the diagonal brace, which is the hypotenuse of a right triangle. Using the Pythagorean theorem (again!), we get 48 squared plus 64 squared equals the length of the diagonal brace squared. Crunching the numbers, we find that the length should be approximately 80 inches. So, the answer is A, because apparently triangles like to give us long diagonal hugs.

4. Oh, Bashir and his weather balloon adventures! Now, we have a height of 4 kilometers and a distance of 1 kilometer from the station. So, we're looking for the distance between the station and the balloon, which is, once again, the hypotenuse of a right triangle. Using the Pythagorean theorem (yes, again and again!), we get 1 squared plus 4 squared equals the distance squared. Doing some quick calculations, we find that the approximate distance is 4.1 kilometers. So, the answer is C, because apparently weather balloons like to keep the station on its toes (or maybe just up in the air).

Great! Here are the step-by-step explanations for the answers to the Converse of Pythagorean Theorem Practice:

1. To determine if a set of numbers represents a Pythagorean triple, you can use the Pythagorean Theorem itself. The theorem states that for any 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. So, let's check if the numbers 27, 36, and 45 satisfy this condition:
- Length of hypotenuse (45) squared = 2025
- Sum of the squares of the other two sides (27^2 + 36^2) = 2025
Since both sides of the equation are equal, we can conclude that 27, 36, and 45 represent a Pythagorean triple. The correct answer is D.

2. To find the approximate length of the hypotenuse, we can use the Pythagorean Theorem. The theorem states that for any 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. So, using the given information:
- Length of one leg = 7
- Length of the other leg = 8
- Length of hypotenuse squared = 7^2 + 8^2 = 49 + 64 = 113
- Length of hypotenuse ≈ √113 ≈ 10.6 (rounded to one decimal place)
Therefore, the approximate length of the hypotenuse is 10.6. The correct answer is D.

3. To find the length of the diagonal brace, we can use the Pythagorean Theorem. The theorem states that for any 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:
- Height of the post = 48 inches
- Distance between the posts = 64 inches
- Length of the diagonal brace (hypotenuse) squared = 48^2 + 64^2
- Length of the diagonal brace ≈ √(48^2 + 64^2) ≈ 80 (rounded to the nearest inch)
Therefore, the length of the diagonal brace should be approximately 80 inches. The correct answer is A.

4. To find the approximate distance between the station and the balloon, we can use the Pythagorean Theorem. The theorem states that for any 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:
- Height of the balloon = 4 kilometers
- Distance from the station = 1 kilometer
- Distance (hypotenuse) squared = 1^2 + 4^2 = 1 + 16 = 17
- Distance ≈ √17 ≈ 4.1 (rounded to one decimal place)
Therefore, the approximate distance between the station and the balloon is 4.1 kilometers. The correct answer is C.

I hope this helps! If you have any more questions, feel free to ask.

To find the answer for these questions, we need to use the Converse of the Pythagorean Theorem. The converse states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

1. To determine which set of numbers represents a Pythagorean triple, we need to calculate the squares of each number and check if they satisfy the Pythagorean Theorem. Let's check the given set of numbers:
27^2 + 36^2 = 729 + 1296 = 2025
45^2 = 2025

Since the sum of the squares of the shorter sides (27 and 36) is equal to the square of the longest side (45), this set of numbers represents a Pythagorean triple.

2. To find the approximate length of the hypotenuse, we can use the Pythagorean Theorem. Let's plug in the given values: a = 7, b = 8, and c (hypotenuse) is what we need to find.
a^2 + b^2 = c^2
7^2 + 8^2 = c^2
49 + 64 = c^2
113 = c^2

To find the approximate length of the hypotenuse, we can take the square root of both sides:
√113 ≈ 10.6

Therefore, the approximate length of the hypotenuse is 10.6.

3. To find the length of the diagonal brace, we can again use the Pythagorean Theorem. The height of the post represents one side, the distance between the posts represents another side, and the length of the diagonal brace represents the hypotenuse. Let's plug in the given values: a = 48, b = 64, and c (diagonal brace) is what we need to find.
a^2 + b^2 = c^2
48^2 + 64^2 = c^2
2304 + 4096 = c^2
6400 = c^2

To find the length of the diagonal brace, we can take the square root of both sides:
√6400 = 80

Therefore, the length of the diagonal brace should be 80 inches.

4. To determine the approximate distance between the station and the balloon, we can again use the Pythagorean Theorem. The height of the balloon represents one side, the distance from the station represents another side, and the distance between the station and the balloon represents the hypotenuse. Let's plug in the given values: a = 4, b = 1, and c (distance between the station and balloon) is what we need to find.
a^2 + b^2 = c^2
4^2 + 1^2 = c^2
16 + 1 = c^2
17 = c^2

To find the approximate distance between the station and the balloon, we can take the square root of both sides:
√17 ≈ 4.1

Therefore, the approximate distance between the station and the balloon is 4.1 kilometers.