Evaluate the lengths of an altitude and the missing sides using the cosine and the tangent.
Perimeter: Approximately 8.74ft Area: Approximately 3.4ft^2
Practice makes perfect
We are asked to find the perimeter and the area of the given isosceles triangle. To do this, we need to find the lengths of an altitude and the missing sides, which we will call x and y. Remember that an altitude is always perpendicular to the base.
Recall that in an isosceles triangle the altitude divides the base into two congruent segments. As we can see, the altitude divided the triangle into two right triangles.
Since we are given an angle measure of a right triangle, we can use the trigonometric ratios to evaluate the values of x and y. Let's begin with recalling 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 ∠ 48^(∘) is a ratio of the length of the leg adjacent to ∠ 48^(∘), which is 1.75, to the length of the hypotenuse, x.
cos 48^(∘)=1.75/x
Next, we will solve the equation.
The value of x is approximately 2.62. To find the value of y, let's recall 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 y. The leg opposite ∠ 48^(∘) is y and the leg adjacent ∠ 48^(∘) is 1.75.
tan 48^(∘)=y/1.75
Let's solve this 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: 2.62+ 2.62+3.5=8.74
The perimeter of the triangle is approximately 8.74 feet. Next, let's recall that the area of a triangle is the half of the product of its base and corresponding altitude.
Area: 1/2*3.5* 1.94≈ 3.4
The area of the triangle is approximately 3.4 square feet.
