# Find the Side Length of A Right Triangle

Sohcahtoa and the Pythagorean Theorem

There are many ways to find the side length of a right triangle. We are going to focus on two specific cases.

Case I

When we know 2 sides of the right triangle, use the Pythagorean theorem.

Case II

We know 1 side and 1 angle of the right triangle, in which case, use sohcahtoa.

### Practice Problems

Calculate the length of the sides below. In each case, round your answer to the nearest hundredth.

##### Problem 1
Step 1

Since we know 2 sides of this triangle, we will use the Pythagorean theorem to solve for x.

Step 2

Substitute the two known sides into the Pythagorean theorem's formula:

$$a^2 + b^2 = c^2 \\ 8^2 + 6^2 = x^2 \\ 100 = x^2 \\ x = \sqrt{100} \\ x = \boxed{10}$$

##### Problem 2
Step 1

Since we know 1 side and 1 angle of this triangle, we will use sohcahtoa.

Step 2

Set up an equation using a sohcahtoa ratio. Since we know the hypotenuse and want to find the side opposite of the 53° angle, we are dealing with sine

$$sin(53) = \frac{ opposite}{hypotenuse} \\ sin(53) = \frac{ \red x }{ 12 }$$

Now, just solve the Equation:

Step 3

$$sin(53) = \frac{ \red x }{ 12 } \\ \red x = 12 \cdot sin (53) \\ \red x = \boxed{ 11.98}$$

##### Problem 3
Step 1

Since we know 2 sides of this triangle, we will use the Pythagorean theorem to solve for side t.

Step 2

Substitute the two known sides into the Pythagorean theorem's formula:

$$a^2 + b^2 = c^2 \\ \red t^2 + 12^2 = 13^2 \\ \red t^2 + 144 = 169 \\ \red t^2 = 169 - 144 \\ \red t^2 = 25 \\ \red t = \boxed{5}$$

##### Problem 4
Step 1

Since we know 1 side and 1 angle of this triangle, we will use sohcahtoa.

Step 2

Set up an equation using the sine, cosine or tangent ratio Since we want to know the length of the hypotenuse, and we already know the side opposite of the 53° angle, we are dealing with sine.

$$sin(67) = \frac{opp}{hyp} \\ sin(67) = \frac{24}{\red x}$$

Now, just solve the Equation:

Step 3

$$x = \frac{ 24}{ sin(67) } \\ x = 26.07$$

##### Problem 5
Step 1

Since we know 2 sides and 1 angle of this triangle, we can use either the Pythagorean theorem (by making use of the two sides) or use sohcahtoa (by making use of the angle and 1 of the given sides).

Step 2

Chose which way you want to solve this problem. There are several different solutions. The only thing you cannot use is sine, since the sine ratio does not involve the adjacent side, x, which we are trying to find.

Pythagorean Theorem Using Cosine Using Tangent

A² + B² = C²

The answers are slightly different (tangent s 35.34 vs 36 for the others) due to rounding issues. I rounded the angle's measure to 23° for the sake of simplicity of the diagram. A more accurate angle measure would have been 22.61986495°. If you use that value instead of 23°, you will get answers that are more consistent.

Step 3

$$x = \frac{ 24}{ sin(67) } \approx 26.07$$