Question

A right triangle can be formed using three sides having lengths 18 inches, 24 inches, and

12
15
16
28
30

Answer 1

To determine which of the given lengths can form a right triangle with the sides 18 inches and 24 inches, we can use the Pythagorean theorem. In a right triangle, the relationship between the lengths of the sides \( a \), \( b \), and the hypotenuse \( c \) is given by:

\[
a^2 + b^2 = c^2
\]

Assuming 24 inches is the longest side (hypotenuse), we have:

\[
18^2 + x^2 = 24^2
\]

Calculating the squares:

\[
324 + x^2 = 576
\]

Subtracting 324 from both sides:

\[
x^2 = 576 - 324 = 252
\]

Taking the square root:

\[
x = \sqrt{252} \approx 15.5
\]

So, approximately \( x \approx 15.5 \) inches can be a side that completes a right triangle with the other two lengths.

Now let's check the provided lengths against this calculation:

- 12 (too short)
- 15 (could work)
- 16 (too long)
- 28 (too long)
- 30 (too long)

To verify, let's check if any of the provided lengths work as potential hypotenuses:

1. **15 inches as hypotenuse:**
\[
18^2 + 24^2 = 324 + 576 = 900 \Rightarrow \sqrt{900} = 30 \quad (\text{no match})
\]

2. **12 inches as hypotenuse:**
\[
18^2 + 12^2 = 324 + 144 = 468 \quad (\text{no match})
\]

3. **16 inches as hypotenuse:**
\[
18^2 + 16^2 = 324 + 256 = 580 \quad (\text{no match})
\]

4. **28 inches as hypotenuse:**
\[
18^2 + 24^2 = 324 + 576 = 900 \Rightarrow \sqrt{900} = 30 \quad (\text{no match})
\]

5. **30 inches as hypotenuse:**
\[
18^2 + 24^2 = 324 + 576 = 900 \Rightarrow 30^2 = 900 \quad (match!)
\]

Therefore, the length **30 inches** can form a right triangle with the sides 18 inches and 24 inches.

Thus, the correct option is **30**.