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 \( \angle ABC \) is the right angle, the length of \( AC \) is given, and \( \angle ACB \) 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 \( \angle ACB \), we will use trigonometric identities.
- Using the cosine function to find \( AB \):
\[ \cos(\theta) = \frac{BC}{AC} \implies BC = AC \cdot \cos(\theta) \]
- Using the sine function to find \( BC \):
\[ \sin(\theta) = \frac{AB}{AC} \implies AB = AC \cdot \sin(\theta) \]
Step-by-Step Calculation
Let's go through an example to illustrate how to use these formulas.
Example:
Given:
- \( AC = 10 \) units (the hypotenuse)
- \( \angle ACB = 30^\circ \)
We want to find the perimeter of the triangle.
Step 1: Identify the Given Values
Given:
- \( AC = 10 \) units
- \( \angle ACB = 30^\circ \)
Step 2: Use Trigonometric Functions to Find the Other Sides
1. Find \( BC \) (the adjacent side to \( \angle ACB \)):
\[\cos(30^\circ) = \frac{BC}{10} \implies BC = 10 \cdot \cos(30^\circ)\]
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 \( \angle ACB \)):
\[\sin(30^\circ) = \frac{AB}{10} \implies AB = 10 \cdot \sin(30^\circ)\]
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 \( \angle ACB = 30^\circ \) 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.
