How To Find The Perimeter Of A Right Triangle ABC Where The Length Of AC And The Angle ACB Are Given

Calculating the perimeter of a right triangle when you know one side and one non-right angle involves using trigonometric functions to find the lengths of the other sides. Here's a step-by-step guide to determine the perimeter of right triangle ABC where $\mathrm{\angle }ABC$ is the right angle, the length of $AC$ is given, and $\mathrm{\angle }ACB$ is known.

Formula to Find the Perimeter of a Right Triangle

The perimeter $P$ of a right triangle ABC can be calculated using the following formula:

$$P=AB+BC+AC$$

Where:

- $P$ is the perimeter of the triangle.

- $AB$, $BC$, and $AC$ are the lengths of the sides of the triangle.

Explanation of the Formulas

To find the lengths of the sides $AB$ and $BC$ using the given $AC$ (hypotenuse) and $\mathrm{\angle }ACB$, we will use trigonometric identities.

- Using the cosine function to find $AB$:

 $$\cos (\theta )=\frac{BC}{AC}⟹BC=AC\cdot \cos (\theta )$$

- Using the sine function to find $BC$:

 $$\sin (\theta )=\frac{AB}{AC}⟹AB=AC\cdot \sin (\theta )$$

Step-by-Step Calculation

Let's go through an example to illustrate how to use these formulas.

Example:

Given:

- $AC=10$ units (the hypotenuse)

- $\mathrm{\angle }ACB={30}^{\circ }$

We want to find the perimeter of the triangle.

Step 1: Identify the Given Values

Given:

- $AC=10$ units

- $\mathrm{\angle }ACB={30}^{\circ }$

Step 2: Use Trigonometric Functions to Find the Other Sides

1. Find $BC$ (the adjacent side to $\mathrm{\angle }ACB$):

$$\cos ({30}^{\circ })=\frac{BC}{10}⟹BC=10\cdot \cos ({30}^{\circ })$$

Using $\cos ({30}^{\circ })=\frac{\sqrt{3}}{2}$:

$$BC=10\cdot \frac{\sqrt{3}}{2}=5\sqrt{3}\approx 8.66\text{ units}$$

2. Find $AB$ (the opposite side to $\mathrm{\angle }ACB$):

$$\sin ({30}^{\circ })=\frac{AB}{10}⟹AB=10\cdot \sin ({30}^{\circ })$$

Using $\sin ({30}^{\circ })=\frac{1}{2}$:

$$AB=10\cdot \frac{1}{2}=5\text{ units}$$

Step 3: Calculate the Perimeter

Now that we have all three sides, we can find the perimeter:

$$P=AB+BC+AC$$

Substitute the values:

$$P=5+8.66+10$$

Step 4: Calculate the Final Value

$$P\approx 23.66\text{ units}$$

Final Value

The perimeter of the right triangle ABC with $AC=10$ units and $\mathrm{\angle }ACB={30}^{\circ }$ is approximately 23.66 units. 

Using trigonometric functions to find the unknown sides makes it straightforward to calculate the perimeter of a right triangle when given one side and one non-right angle.