Question

which side lengths form a right triangle?
choose all answers that apply:

1, 5, 26
1.5, 2, 2.5
4, 4, √32

Answer 1

To determine which sets of side lengths form a right triangle, we can use the Pythagorean theorem, which states that for a triangle with side lengths \(a\), \(b\), and hypotenuse \(c\) (where \(c\) is the longest side), the following must hold:

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

Now, let's examine each set of side lengths:

1. **Side lengths: 1, 5, 26**
- Here, \(c = 26\), \(a = 1\), and \(b = 5\).
- Calculate:
\[
1^2 + 5^2 = 1 + 25 = 26
\]
\[
c^2 = 26^2 = 676
\]
\[
1 + 25 \neq 676 \quad \text{(Not a right triangle)}
\]

2. **Side lengths: 1.5, 2, 2.5**
- Here, \(c = 2.5\), \(a = 1.5\), and \(b = 2\).
- Calculate:
\[
1.5^2 + 2^2 = 2.25 + 4 = 6.25
\]
\[
c^2 = (2.5)^2 = 6.25
\]
\[
2.25 + 4 = 6.25 \quad \text{(This is a right triangle)}
\]

3. **Side lengths: 4, 4, √32**
- Here, \(c = \sqrt{32}\), and \(a = 4\), \(b = 4\).
- Calculate:
\[
4^2 + 4^2 = 16 + 16 = 32
\]
\[
c^2 = (\sqrt{32})^2 = 32
\]
\[
16 + 16 = 32 \quad \text{(This is a right triangle)}
\]

Based on the Pythagorean theorem, the sets that form a right triangle are:

- \(1.5, 2, 2.5\)
- \(4, 4, \sqrt{32}\)

So the correct answers are:
- 1.5, 2, 2.5
- 4, 4, √32