Pythagoras & Trigonometry - Free Printable

Problem Description:

The task is to match the side lengths of right-angled triangles in the given diagrams with the provided options. Each card (side length) must be used exactly once. The diagrams are not drawn to scale, so we need to use trigonometric relationships and properties of special right triangles to determine the correct side lengths.

Step-by-Step Solution:

#### 1. Understanding the Diagrams:
We have six right-angled triangles, each with specific angles marked. The goal is to determine the side lengths using the given options:
- \(1\)
- \(\sqrt{2}\)
- \(\sqrt{3}\)
- \(2\)
- \(3\)

#### 2. Analyzing Each Triangle:

##### Triangle 1: \(30^\circ\)
- This is a 30-60-90 triangle.
- In a 30-60-90 triangle, the side lengths are in the ratio:
\[
\text{Opposite } 30^\circ : \text{Opposite } 60^\circ : \text{Hypotenuse} = 1 : \sqrt{3} : 2
\]
- Let's assume the shortest side (opposite \(30^\circ\)) is \(1\). Then:
- The side opposite \(60^\circ\) is \(\sqrt{3}\).
- The hypotenuse is \(2\).

##### Triangle 2: \(45^\circ\)
- This is a 45-45-90 triangle.
- In a 45-45-90 triangle, the side lengths are in the ratio:
\[
\text{Leg} : \text{Leg} : \text{Hypotenuse} = 1 : 1 : \sqrt{2}
\]
- Let's assume each leg is \(1\). Then:
- The hypotenuse is \(\sqrt{2}\).

##### Triangle 3: \(60^\circ\)
- This is another 30-60-90 triangle.
- Using the same ratio as before:
\[
\text{Opposite } 30^\circ : \text{Opposite } 60^\circ : \text{Hypotenuse} = 1 : \sqrt{3} : 2
\]
- However, this time, let's assume the side opposite \(60^\circ\) is \(\sqrt{3}\). Then:
- The side opposite \(30^\circ\) is \(1\).
- The hypotenuse is \(2\).

##### Triangle 4: \(30^\circ\)
- This is another 30-60-90 triangle.
- Using the same ratio:
\[
\text{Opposite } 30^\circ : \text{Opposite } 60^\circ : \text{Hypotenuse} = 1 : \sqrt{3} : 2
\]
- Let's assume the hypotenuse is \(2\). Then:
- The side opposite \(30^\circ\) is \(1\).
- The side opposite \(60^\circ\) is \(\sqrt{3}\).

##### Triangle 5: \(45^\circ\)
- This is another 45-45-90 triangle.
- Using the same ratio:
\[
\text{Leg} : \text{Leg} : \text{Hypotenuse} = 1 : 1 : \sqrt{2}
\]
- Let's assume each leg is \(1\). Then:
- The hypotenuse is \(\sqrt{2}\).

##### Triangle 6: \(50^\circ\)
- This is a general right triangle with no specific angle ratio.
- However, since we are limited to the given options and must use each option exactly once, we can infer that the side lengths must fit the remaining options.
- The only remaining option is \(3\). Since there are no other constraints, we can assign \(3\) to one of the sides.

#### 3. Matching the Side Lengths:
Now, let's match the side lengths to the triangles:

- Triangle 1 (30°): \(1, \sqrt{3}, 2\)
- Triangle 2 (45°): \(1, 1, \sqrt{2}\)
- Triangle 3 (60°): \(1, \sqrt{3}, 2\)
- Triangle 4 (30°): \(1, \sqrt{3}, 2\)
- Triangle 5 (45°): \(1, 1, \sqrt{2}\)
- Triangle 6 (50°): \(3\) (remaining option)

#### 4. Final Answer:
The side lengths for each triangle are:
- Triangle 1: \(1, \sqrt{3}, 2\)
- Triangle 2: \(1, 1, \sqrt{2}\)
- Triangle 3: \(1, \sqrt{3}, 2\)
- Triangle 4: \(1, \sqrt{3}, 2\)
- Triangle 5: \(1, 1, \sqrt{2}\)
- Triangle 6: \(3\)

Thus, the final answer is:
\[
\boxed{1, \sqrt{3}, 2, 1, 1, \sqrt{2}, 1, \sqrt{3}, 2, 1, \sqrt{3}, 2, 1, 1, \sqrt{2}, 3}
\]