First evaluate the length of the hypotenuse of the old garden. Then recall the Law of Sines.
≈ 96 ft
Practice makes perfect
We are given that Crystal has an organic vegetable garden that is a right triangle, and we know that she wants to add another triangular section. Our task is to evaluate the perimeter of the new garden, so let's take a look at the given picture.
First, we can evaluate the length of the hypotenuse of the old garden, which we will call h. To do this, we will use the Pythagorean Theorem. Recall that, according to this theorem, the sum of the squared legs in a triangle is equal to its squared hypotenuse.
20^2+ 18^2= h^2
Let's solve the above equation. Notice that, since h represents the side length, we will consider only the positive case when taking a square root of h^2.
Next we will focus on the new garden. First we will find the measure of the missing angle. Let's call it x.
To do this, we will use the Triangle Angle Sum Theorem.
x+52^(∘)+74^(∘)=180^(∘) ⇓
x=54^(∘)
The missing angle has a measure of 54^(∘). Since we are asked to find the perimeter of the new garden, we need to know all the side lengths. Let a and b be the missing sides.
To find the missing values, we will use the Law of Sines. According to this law, in a triangle the ratio between the sine of an angle and the length of the side opposite to this angle is constant. Using this, we can write a proportion.
sin 52^(∘)/a=sin 54^(∘)/26.9=sin74^(∘)/b
Let's start with evaluating the value of a using cross multiplication.
