The problem you provided involves solving for the missing side of a right triangle using the Pythagorean Theorem, which states:

$a^2 + b^2 = c^2$

Where:

• $a$ and $b$ are the legs of the triangle, and
• $c$ is the hypotenuse (the side opposite the right angle).

Let's go through each triangle:

a. Triangle with sides 24 and 10

Here, 24 and 10 are the legs, and we need to find the hypotenuse ($c$).

$c^2 = 24^2 + 10^2 = 576 + 100 = 676$ $c = \sqrt{676} = 26$

The missing side is 26.

b. Triangle with sides 3 and 4

Here, 3 and 4 are the legs, and we need to find the hypotenuse ($c$).

$c^2 = 3^2 + 4^2 = 9 + 16 = 25$ $c = \sqrt{25} = 5$

The missing side is 5.

c. Triangle with sides 27 (hypotenuse) and 18 (leg)

Here, 27 is the hypotenuse, and we need to find the missing leg ($b$).

$27^2 = 18^2 + b^2$ $729 = 324 + b^2$ $b^2 = 729 - 324 = 405$ $b = \sqrt{405} \approx 20.12$

The missing side is approximately 20.12.

d. Triangle with sides 23 (hypotenuse) and 12 (leg)

Here, 23 is the hypotenuse, and we need to find the missing leg ($b$).

$23^2 = 12^2 + b^2$ $529 = 144 + b^2$ $b^2 = 529 - 144 = 385$ $b = \sqrt{385} \approx 19.62$

The missing side is approximately 19.62.

Would you like more details on any of these solutions?

Follow-up questions:

1. How is the Pythagorean Theorem applied in three-dimensional problems?
2. Can the Pythagorean Theorem be used for triangles that aren't right triangles?
3. What are some real-life applications of the Pythagorean Theorem?
4. How would the Pythagorean Theorem change in non-Euclidean geometry?
5. Can you find the hypotenuse if you know only one leg and the area of the triangle?

Tip:

Always check whether you are dealing with the hypotenuse or the legs when solving for the missing side to avoid confusion.