Plot the given points in the coordinate plane. Then evaluate the side lengths using the Distance Formula.
Angle Measure: approximately 82^(∘) Explanation: see solution.
Practice makes perfect
We are given the coordinates of ∠ ABC and asked to evaluate the measure of the largest angle. First, let's plot these points in the coordinate plane and connect them to form a triangle.
Our first step will be to evaluate the side lengths of this triangle. To do this, we will use the Distance Formula. Let's start with AB.
We will evaluate the rest of the side lengths in the same way.
Segment
Points
Distance Formula
Simplify
AB
( -3, 6) & ( 4, 2)
sqrt(( 4-( -3))^2+( 2- 6)^2)
sqrt(65)
BC
( 4, 2) & ( -5, 1)
sqrt(( -5- 4)^2+( 1- 2)^2)
sqrt(82)
AC
( -5, 1) & ( -3, 6)
sqrt(( -3-( -5))^2+( 6- 1)^2)
sqrt(29)
Let's add these lengths to our graph.
Notice that the largest angle in a triangle lies opposite the largest side of this triangle. Therefore, in △ ABC, the largest angle is ∠ CAB. Since we have all three side lengths, we can use the Law of Cosines to evaluate the measure of ∠ CAB. Let's recall this law.
If△ ABC has lengths of a, b,and c and angle measures of A, B,and C, then we can write equations that relate the side lengths of this triangle and the cosine of one of its angles.
Using this law, we will write and solve an equation for ∠ CAB.
Next we can use the inverse cosine to find the measure of ∠ CAB.
cos ∠ CAB=6/sqrt(1885) ⇓
∠ CAB=cos ^(-1)6/sqrt(1885)≈ 82^(∘)
The measure of the largest angle in △ ABC is approximately 82^(∘).
