Pythagoras & Trigonometry - Free Printable

Problem Description:

The task is to match the side lengths of right-angled triangles using the Pythagorean theorem. The given triangles have one hypotenuse and two unknown sides, and we need to determine the missing side lengths using the provided options. Each number from the list on the right can be used only once.

Pythagorean Theorem:

The Pythagorean theorem states:
\[
a^2 + b^2 = c^2
\]
where \( c \) is the hypotenuse (the longest side), and \( a \) and \( b \) are the other two sides.

Step-by-Step Solution:

#### Triangle 1:
- Hypotenuse: \( 10 \)
- One side: Unknown
- Other side: Unknown

We need to find two numbers \( a \) and \( b \) such that:
\[
a^2 + b^2 = 10^2 = 100
\]

From the options, let's try combinations:
- \( 6^2 + 8^2 = 36 + 64 = 100 \)

So, the sides are \( 6 \) and \( 8 \).

#### Triangle 2:
- Hypotenuse: \( 13 \)
- One side: Unknown
- Other side: Unknown

We need to find two numbers \( a \) and \( b \) such that:
\[
a^2 + b^2 = 13^2 = 169
\]

From the options, let's try combinations:
- \( 5^2 + 12^2 = 25 + 144 = 169 \)

So, the sides are \( 5 \) and \( 12 \).

#### Triangle 3:
- Hypotenuse: \( 3 \)
- One side: Unknown
- Other side: Unknown

We need to find two numbers \( a \) and \( b \) such that:
\[
a^2 + b^2 = 3^2 = 9
\]

From the options, let's try combinations:
- \( 1^2 + 2^2 = 1 + 4 = 5 \) (not valid)
- \( 2^2 + 2^2 = 4 + 4 = 8 \) (not valid)

Since none of the options work directly, let's recheck the problem. It seems there might be a mistake in the options or the triangle dimensions. However, if we assume the triangle is not standard, we can use the closest possible values. But for now, let's proceed with the next triangles.

#### Triangle 4:
- Hypotenuse: \( 10 \)
- One side: Unknown
- Other side: Unknown

We need to find two numbers \( a \) and \( b \) such that:
\[
a^2 + b^2 = 10^2 = 100
\]

From the options, let's try combinations:
- \( 6^2 + 8^2 = 36 + 64 = 100 \)

So, the sides are \( 6 \) and \( 8 \).

#### Triangle 5:
- Hypotenuse: \( 25 \)
- One side: Unknown
- Other side: Unknown

We need to find two numbers \( a \) and \( b \) such that:
\[
a^2 + b^2 = 25^2 = 625
\]

From the options, let's try combinations:
- \( 7^2 + 24^2 = 49 + 576 = 625 \)

So, the sides are \( 7 \) and \( 24 \).

#### Triangle 6:
- Hypotenuse: \( 17 \)
- One side: Unknown
- Other side: Unknown

We need to find two numbers \( a \) and \( b \) such that:
\[
a^2 + b^2 = 17^2 = 289
\]

From the options, let's try combinations:
- \( 8^2 + 15^2 = 64 + 225 = 289 \)

So, the sides are \( 8 \) and \( 15 \).

Final Matching:

- Triangle 1: \( 6 \) and \( 8 \)
- Triangle 2: \( 5 \) and \( 12 \)
- Triangle 3: Not solvable with given options
- Triangle 4: \( 6 \) and \( 8 \)
- Triangle 5: \( 7 \) and \( 24 \)
- Triangle 6: \( 8 \) and \( 15 \)

Answer:

\[
\boxed{6, 8, 5, 12, 7, 24, 8, 15}
\]