Given the following sets of side lengths for triangles, select those which would be right-angled:

Select one or more:

a.
13,6,12


b.
100,100,200−−−√


c.
1.5,2.5,2


d.
13−−√,6,7


e.
10,24,26

To determine whether a triangle is right-angled, we can apply the **Pythagorean Theorem**, which states:

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

where \(a\), \(b\), and \(c\) are the side lengths of the triangle, with \(c\) being the hypotenuse (the longest side).

Now, let's check each option:

### a. \( 13, 6, 12 \)
Hypotenuse: \( 13 \)
Checking the Pythagorean Theorem:
\[
6^2 + 12^2 = 13^2 \\
36 + 144 = 169 \\
180 \neq 169
\]
This is **not** a right-angled triangle.

### b. \( 100, 100, \sqrt{200} \)
Hypotenuse: \( \sqrt{200} \)
Checking the Pythagorean Theorem:
\[
100^2 + 100^2 = (\sqrt{200})^2 \\
10000 + 10000 = 200 \\
20000 \neq 200
\]
This is **not** a right-angled triangle.

### c. \( 1.5, 2.5, 2 \)
Hypotenuse: \( 2.5 \)
Checking the Pythagorean Theorem:
\[
1.5^2 + 2^2 = 2.5^2 \\
2.25 + 4 = 6.25 \\
6.25 = 6.25
\]
This **is** a right-angled triangle.

### d. \( \sqrt{13}, 6, 7 \)
Hypotenuse: \( 7 \)
Checking the Pythagorean Theorem:
\[
(\sqrt{13})^2 + 6^2 = 7^2 \\
13 + 36 = 49 \\
49 = 49
\]
This **is** a right-angled triangle.

### e. \( 10, 24, 26 \)
Hypotenuse: \( 26 \)
Checking the Pythagorean Theorem:
\[
10^2 + 24^2 = 26^2 \\
100 + 576 = 676 \\
676 = 676
\]
This **is** a right-angled triangle.

### Conclusion
The sets of side lengths that form right-angled triangles are:
- **c. \(1.5, 2.5, 2\)**
- **d. \( \sqrt{13}, 6, 7 \)**
- **e. \( 10, 24, 26 \)**

Would you like more details or have any questions?

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Here are 5 related questions:
1. How can you derive the hypotenuse given two side lengths?
2. What is the general form of the Pythagorean Theorem?
3. How does the Pythagorean Theorem apply to non-right triangles?
4. What other methods can determine triangle types besides the Pythagorean Theorem?
5. Can a triangle with rational sides always be a right triangle?

**Tip:** When using the Pythagorean Theorem, always ensure you identify the correct hypotenuse for accurate calculations.

Discover how to identify right-angled triangles using the Pythagorean Theorem. Check specific sets of side lengths and determine which triangles are right-angled. Includes detailed examples and explanations.

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