A triangular plot of farm land measures 0.9 by 0.5 by 1.25 miles. We want to find the angles between each side of the fence situated on the borders of the land. Let's consider this on a diagram. We will name each corner of the fence.
Let's find the angle measures one at a time.
Finding m∠B
We will start with finding the measure of the largest angle, ∠B. To do so we can use the Law of Cosines.
Let's substitute a= 0.5, b= 1.25, and c= 0.9 to isolate cos B.
Finally, to find m ∠A we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180.
m ∠A+ 124+ 37 = 180 ⇔ m ∠A ≈ 19
Finally, we have that the angles at which the fences of the three sides meet are 124^(∘), 37^(∘) and 19^(∘).
b
We will now find the area of the plot of land. Let's add the angle measures to our diagram.
To find the desired area we will use the formula for finding the area of a triangle using sine.
Area=1/2ac sin B
In our triangle we have a= 0.5, c= 0.9, and m∠B≈ 124^(∘). We can substitute these values into the formula to find the area. Let's do it!
