Let's go through the questions and problems step by step.

Part A: Stop and Think Questions

These questions involve finding missing side lengths of right triangles. We'll use the Pythagorean Theorem:
$a^2 + b^2 = c^2$ where $a$ and $b$ are the legs, and $c$ is the hypotenuse of a right triangle.

1. What is the length of the hypotenuse in a right triangle with one leg of length 3 and the other leg 4?

   • Using the Pythagorean Theorem:
     $3^2 + 4^2 = c^2 \implies 9 + 16 = c^2 \implies c^2 = 25 \implies c = 5$ The hypotenuse is 5.

2. What is the length of the hypotenuse in a right triangle with one leg of length 7 and the other leg 24?

   • Using the Pythagorean Theorem:
     $7^2 + 24^2 = c^2 \implies 49 + 576 = c^2 \implies c^2 = 625 \implies c = 25$ The hypotenuse is 25.

3. What is the length of one leg in a right triangle which has the other leg of length 8 and the hypotenuse 10?

   • Using the Pythagorean Theorem:
     $a^2 + 8^2 = 10^2 \implies a^2 + 64 = 100 \implies a^2 = 36 \implies a = 6$ The missing leg is 6.

4. What is the length of one leg in a right triangle which has the other leg of length 16 and the hypotenuse 20?

   • Using the Pythagorean Theorem:
     $a^2 + 16^2 = 20^2 \implies a^2 + 256 = 400 \implies a^2 = 144 \implies a = 12$ The missing leg is 12.

Part B: Determine the missing side lengths in the right triangles below

5. Right ∠Y (Triangle XYZ)

   • We are given $XY = 10$ and $XZ = 15$, and we need to find $YZ$.
   • Using the Pythagorean Theorem:
     $10^2 + YZ^2 = 15^2 \implies 100 + YZ^2 = 225 \implies YZ^2 = 125 \implies YZ = \sqrt{125} = 5\sqrt{5}$ The length of $YZ$ is 5√5.

6. Right ∠C (Triangle ABC)

   • We are given $AB = 50$ (the hypotenuse) and $BC = 40$, and we need to find $AC$.
   • Using the Pythagorean Theorem:
     $AC^2 + 40^2 = 50^2 \implies AC^2 + 1600 = 2500 \implies AC^2 = 900 \implies AC = 30$ The length of $AC$ is 30.

7. Right ∠C (Triangle ABC)

   • We are given $AB = 5$ and $BC = 5$, and we need to find $AC$.
   • Using the Pythagorean Theorem:
     $5^2 + 5^2 = AC^2 \implies 25 + 25 = AC^2 \implies AC^2 = 50 \implies AC = \sqrt{50} = 5\sqrt{2}$ The length of $AC$ is 5√2.

Would you like any further explanation for any of these steps?

Here are some related questions to further your understanding:

1. What happens if the hypotenuse and both legs are all integers, like in a "Pythagorean triple"?
2. Can you find the hypotenuse if both legs are given as square roots?
3. How does the Pythagorean Theorem apply to real-world problems involving distances?
4. Could you explain how to find the length of the third side of a triangle if you only know the hypotenuse and one leg?
5. How can you simplify radical expressions like √50 into simpler forms?

Tip: Always check if you can simplify square roots (e.g., √50 = 5√2) for your final answer.