Let's start solving the problems shown in your image.

Question 1:

You need to find the missing side $x$ of a right triangle where one leg is 10, the hypotenuse is 12.7, and the other leg is $x$.

To solve this, we can use the Pythagorean Theorem: $a^2 + b^2 = c^2$ Where:

• $a = 10$ (one leg),
• $b = x$ (the missing side),
• $c = 12.7$ (the hypotenuse).

So, $10^2 + x^2 = 12.7^2$ $100 + x^2 = 161.29$ $x^2 = 161.29 - 100 = 61.29$ $x = \sqrt{61.29} \approx 7.83$ So, $x \approx 7.8$ (rounded to the nearest tenth).

Question 12:

You need to find the hypotenuse of a right triangle where the legs are 21 units and 28 units.

Again, using the Pythagorean Theorem: $a^2 + b^2 = c^2$ Where:

• $a = 21$,
• $b = 28$,
• $c =$ hypotenuse.

$21^2 + 28^2 = c^2$ $441 + 784 = c^2$ $c^2 = 1225$ $c = \sqrt{1225} = 35$ So, the hypotenuse is 35 units.

Question 13:

You need to find the hypotenuse of a right triangle where one leg is 7 units and the other leg is 24 units.

Using the Pythagorean Theorem: $7^2 + 24^2 = c^2$ $49 + 576 = c^2$ $c^2 = 625$ $c = \sqrt{625} = 25$ So, the hypotenuse is 25 units.

Would you like more details on any of these solutions, or do you have any questions?

Here are 5 related questions you can think about:

1. How does the Pythagorean theorem apply to different types of triangles?
2. What happens when the triangle has sides that don't meet the Pythagorean relationship?
3. How can you calculate the angles of a right triangle if you know all the sides?
4. Can the Pythagorean theorem be used for non-right triangles?
5. How would the solution change if one side was irrational?

Tip: Always check that your triangle is a right triangle before applying the Pythagorean theorem!