Answer:
In any triangle, the sum of the angles is always 180 degrees. So we can use this fact to find the measure of angle A and angle B:
m∠C + m∠A + m∠B = 180
Substituting the given values, we get:
97° + 47° + 36° = 180°
Simplifying this equation, we get:
180° = 180°
This confirms that the given angle measures are valid.
To find the largest side of the triangle, we can use the law of sines, which relates the sides of a triangle to the sines of their opposite angles:
a/sin(A) = b/sin(B) = c/sin(C)
where a, b, and c are the lengths of the sides opposite angles A, B, and C, respectively.
We want to find the largest side, which is opposite the largest angle. Since angle C is the largest angle, we can set up the proportion:
a/sin(A) = b/sin(B) = c/sin(C)
a/sin(47°) = b/sin(36°) = c/sin(97°)
To find the largest side, we need to find the largest value of the expression a/sin(47°). We can start by finding the value of b/sin(36°) and c/sin(97°) using the given information.
Using the law of sines, we can write:
b/sin(36°) = c/sin(97°)
b = (sin(36°)/sin(97°))c
Now, substituting this expression for b, we get:
a/sin(47°) = (sin(36°)/sin(97°))c/sin(36°)
Simplifying this expression, we get:
a/sin(47°) = c/sin(97°)
a = (sin(47°)/sin(97°))c
Now, we need to find the value of c. Using the law of sines again, we can write:
a/sin(47°) = b/sin(36°) = c/sin(97°)
c = (sin(97°)/sin(47°))a
Substituting the expression we found for a, we get:
c = (sin(97°)/sin(47°))(sin(47°)/sin(97°))c
c = c
So, we see that c is equal to itself, which doesn't give us any new information. However, we can use the expression we found for a to determine the largest side:
a = (sin(47°)/sin(97°))c
Since sin(47°) < sin(90°) and sin(97°) > sin(90°), we know that sin(47°)/sin(97°) < 1. Therefore, the largest side of the triangle is c, which is opposite the largest angle, angle C.
In conclusion, we cannot determine the exact length of the largest side without additional information, but we know that it is opposite angle C, which has a measure of 97 degrees.