Question

Find the largest side of △ABC, given that m∠C=97°, m∠A=47°, and m∠B=36°.

Answer 1

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.

Answer 2

We can use the Law of Sines to solve for the sides of the triangle. The Law of Sines states that for any triangle with sides a, b, and c opposite angles A, B, and C, respectively:

a/sin A = b/sin B = c/sin C

We can use this formula to solve for any side of a triangle if we know the measures of two angles and the length of one side.

Let's label the sides of the triangle as follows:

a is opposite angle A (which measures 47°)

b is opposite angle B (which measures 36°)

c is opposite angle C (which measures 97°)

We want to find the largest side, so let's solve for all three sides using the Law of Sines and then compare them:

a/sin 47° = b/sin 36° = c/sin 97°

Solving for a:

a = sin 47° (b/sin 36°) = (sin 47°/sin 36°) b

Solving for b:

b = sin 36° (a/sin 47°) = (sin 36°/sin 47°) a

Solving for c:

c = sin 97° (a/sin 47°) = (sin 97°/sin 47°) a

To compare the sides, we can simply compare the coefficients in front of a and b:

a = (sin 47°/sin 36°) b ≈ 1.336 b

c = (sin 97°/sin 47°) a ≈ 2.117 a

Therefore, c is the largest side of the triangle. We can find its exact value by plugging in the given angle measures:

c = (sin 97°/sin 47°) a ≈ 2.117 a

a/sin 47° = c/sin 97°

a = (sin 47°/sin 97°) c ≈ 0.702 c

So the largest side, c, is approximately 2.117 times the length of the smallest side, a. We don't know the actual length of any side, since we don't have any side lengths to use as reference.