Question

You need to find the lengths of two legs of a triangle. You have the hypotenuse length (8) and you have all three angles (90, 30, and 60). How do you do it?

Answer 1

Absolutely, it's a well-known problem and it can be solved by using the properties of a 30-60-90 triangle.

First, let's recall some geometric properties. In a 30-60-90 triangle, the sides are in the ratio of 1:√3:2. Specifically, the length of the side opposite the 30° angle (let's call it side a) is half the length of the hypotenuse, while the length of the side opposite the 60° angle (let's call it side b) is the length of the hypotenuse divided by 2 and then multiplied by √3.

So, knowing this, given that the length of the hypotenuse is 8, we can calculate the lengths of the other two sides.

Let's calculate it for side a first.

We know from the ratio above that side a (opposite the 30° angle) equals half the length of the hypotenuse, so:

```
side_a = hypotenuse / 2
```

And given that the hypotenuse equals 8, we get:

```
side_a = 8 / 2
side_a = 4
```

So, the length of side a (opposite the 30° angle) is 4 units.

Now, let's calculate it for side b:

We also know from the ratio above that side b (opposite the 60° angle) equals the length of the hypotenuse divided by 2 and then multiplied by √3. So, we get:

```
side_b = (hypotenuse / 2) * √3
```

Then, substituting the length of the hypotenuse (8), we get:

```
side_b = (8 / 2) * √3
side_b = 4 * √3
side_b ≈ 6.928203230275509
```

There we go. The lengths of sides a and b are approximately 4 units and 6.93 units, respectively.