Determine the measure of the angle C to the nearest degree. 11 cm 7 cm 38° C
The Correct Answer and Explanation is:
The correct measure of angle C to the nearest degree is 105°.
Here is a step-by-step explanation of how to find the measure of angle C:
1. Identify the Given Information
From the provided image of triangle ABC, we have the following known values:
- The length of the side opposite angle B (side AC) is b = 7 cm.
- The measure of angle B is 38°.
- The length of the side opposite angle C (side AB) is c = 11 cm.
We need to determine the measure of angle C.
2. Choose the Appropriate Trigonometric Law
This problem involves a triangle where we know two sides and a non-included angle (SSA configuration). The Law of Sines is the appropriate tool for this situation. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides. The formula is:
a/sin(A) = b/sin(B) = c/sin(C)
3. Set Up the Equation
Using the parts of the Law of Sines that involve our known and unknown values, we can set up the following proportion:
b / sin(B) = c / sin(C)
Substitute the given values into the equation:
7 / sin(38°) = 11 / sin(C)
4. Solve for Angle C
To solve for angle C, we first need to isolate sin(C). We can do this by cross-multiplying:
7 * sin(C) = 11 * sin(38°)
Next, divide both sides by 7:
sin(C) = (11 * sin(38°)) / 7
Now, use a calculator to find the value of sin(38°) and perform the calculation:
sin(38°) ≈ 0.6157
sin(C) ≈ (11 * 0.6157) / 7
sin(C) ≈ 6.7727 / 7
sin(C) ≈ 0.9675
To find the angle C, we take the inverse sine (arcsin) of this value:
C = arcsin(0.9675)
C ≈ 75.4°
5. Consider the Ambiguous Case (SSA)
The Side-Side-Angle (SSA) case is known as the ambiguous case because it can sometimes result in two possible triangles. When sin(C) is calculated, there are two possible angles between 0° and 180°: an acute angle (C₁) and an obtuse angle (C₂ = 180° – C₁).
- The first possible angle is C₁ ≈ 75.4°.
- The second possible angle is C₂ = 180° – 75.4° = 104.6°.
To determine which angle is correct, we look at the diagram. The drawing clearly shows that angle C is an obtuse angle, meaning it is greater than 90°. Therefore, we must choose the obtuse solution.
6. Final Calculation and Rounding
The correct measure for angle C is approximately 104.6°. The problem asks for the answer to be rounded to the nearest degree.
Rounding 104.6° to the nearest degree gives us 105°.
