Evaluate the lengths of the missing sides using the cosine and the tangent.
Perimeter: Approximately 28.52cm Area: Approximately 23.4cm^2
Practice makes perfect
We are asked to find the perimeter and the area of the given right triangle. To do this, we need to find the lengths of the missing sides. Let's call them x and y.
Since we are given an angle measure of this triangle, we can use the trigonometric ratios to evaluate the lengths of missing sides. Let's begin with recalling the definition of the tangent of an angle.
If△ ABCis a right triangle with acute∠ A,
then the tangent of∠ Ais the ratio of the length of the leg opposite∠ A to the length of the leg adjacent∠ A.
Using this information, we can write an equation to find the value of x. The leg opposite ∠ 18^(∘) is x, and the leg adjacent to ∠ 18^(∘) is 12.
tan 18^(∘)=x/12
Let's solve this equation.
The value of x is approximately 3.9. To find the value of y, let's recall the definition of the cosine of an angle.
If△ ABCis a right triangle with acute∠ A,
then the cosine of∠ Ais the ratio of the length of the leg adjacent∠ A to the length of the hypotenuse.
This means that the cosine of ∠ 18^(∘) is the ratio of the length of the leg adjacent ∠ 18^(∘), which is 12, to the length of the hypotenuse, y.
cos 18^(∘)=12/y
Next, we will solve the equation.
Finally, we will evaluate the perimeter and the area of this triangle. Remember that, since we will use approximate values, the perimeter and the area will also be approximations. First recall that the perimeter of the figure is the sum of all its sides lengths.
Perimeter: 3.9+ 12.62+ 12=28.52
The perimeter of the triangle is approximately 28.52 centimeters. Next, let's recall that the area of a right triangle is the half of the product of its legs.
Area: 1/2* 3.9* 12= 23.4
The area of the triangle is approximately 23.4 square centimeters.
