Question 3
For the triangle shown, calculate the perimeter
2.5 km
51.8°
62.1

The perimeter is 9.31km

Step-by-step explanation:

We have two triangle rectangles, and the perimeter of the complete triangle will be equal to the sum between the hypotenuses (of both triangles) and the bottom cathetus (of both triangles)

For a triangle rectangle and a given angle A (that is not the right angle)

Remember the relationships:

Sin(A) = (opposite cathetus)/(hypotenuse)

Cos(A) = (adjacent cathetus)/(hypotenuse)

Tan(A) = (opposite cathetus)/(adjacent cathetus)

With this, we can find the measures of each triangle in the imae.

The left triangle rectangle.

We have a cathetus (opposite) with length of 2.5km and A = 51.8°

Then the hypotenuse can be found by:

Sin(51.8°) = (2.5km)/(hypotenuse)

then:

hypotenuse = (2.5km)/sin(51.8°) = 3.18 km

And the bottom cathetus can be found with:

Tan(51.8°) = (2.5km)/(adjacent cathetus)

adjacent cathetus =  (2.5km)/(Tan(51.8°)) = 1.98 km

For the other triangle, we have A = 62.1°, and the opposite cathetus is 2.5 km

We can use the same relationships than before:

Sin(62.1°) = (2.5km)/(hypotenuse)

then:

hypotenuse = (2.5km)/sin(62.1°) = 2.83 km

And the bottom cathetus can be found with:

Tan(62.1°) = (2.5km)/(adjacent cathetus)

adjacent cathetus =  (2.5km)/(Tan(62.1°)) = 1.32 km

Then the perimeter of the complete triangle is:

Perimeter = 3.18 km + 1.98 km + 2.83km + 1.32 km = 9.31 km