Answer :
Sure, let's go through a step-by-step solution for the problem:
An isosceles triangle has angle measures 21°, 21°, and 138°. The side across from the 138° angle is 57 cm long. How long are the other sides?
### Step-by-Step Solution:
1. Identify the Triangle's Properties:
- An isosceles triangle has two equal angles.
- Given angles: 21°, 21°, and 138°. The equal angles are both 21°.
- The side opposite the 138° angle (the largest angle) is given as 57 cm.
2. Apply the Law of Sines:
The Law of Sines states:
[tex]\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\][/tex]
Where:
- [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the lengths of the sides of the triangle.
- [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are the measures of the interior angles opposite those sides.
3. Set up the Law of Sines for This Triangle:
Let's denote:
- [tex]\( c = 57 \)[/tex] cm (the side opposite the 138° angle)
- [tex]\( A = 21° \)[/tex], [tex]\( B = 21° \)[/tex], and [tex]\( C = 138° \)[/tex]
According to the Law of Sines:
[tex]\[
\frac{b}{\sin(21°)} = \frac{57}{\sin(138°)}
\][/tex]
4. Calculate the Sine Values:
To perform the calculations, we need the sine values of the angles:
- [tex]\(\sin(21°)\)[/tex]
- [tex]\(\sin(138°)\)[/tex]
Note: [tex]\(\sin(138°) = \sin(180° - 42°) = \sin(42°)\)[/tex], because sin(180° - x) = sin(x).
5. Compute the Other Sides:
[tex]\[
b = \frac{57 \cdot \sin(21°)}{\sin(138°)}
\][/tex]
Simplifying further,
[tex]\[
b = \frac{57 \cdot \sin(21°)}{\sin(42°)}
\][/tex]
6. Evaluate the Expression:
Given the numerical result,
[tex]\[
b = 30.5 \, \text{cm}
\][/tex]
So, the lengths of the other sides of the isosceles triangle are:
C. 30.5 cm
Thus, the correct answer is: C. 30.5 cm.
An isosceles triangle has angle measures 21°, 21°, and 138°. The side across from the 138° angle is 57 cm long. How long are the other sides?
### Step-by-Step Solution:
1. Identify the Triangle's Properties:
- An isosceles triangle has two equal angles.
- Given angles: 21°, 21°, and 138°. The equal angles are both 21°.
- The side opposite the 138° angle (the largest angle) is given as 57 cm.
2. Apply the Law of Sines:
The Law of Sines states:
[tex]\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\][/tex]
Where:
- [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the lengths of the sides of the triangle.
- [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are the measures of the interior angles opposite those sides.
3. Set up the Law of Sines for This Triangle:
Let's denote:
- [tex]\( c = 57 \)[/tex] cm (the side opposite the 138° angle)
- [tex]\( A = 21° \)[/tex], [tex]\( B = 21° \)[/tex], and [tex]\( C = 138° \)[/tex]
According to the Law of Sines:
[tex]\[
\frac{b}{\sin(21°)} = \frac{57}{\sin(138°)}
\][/tex]
4. Calculate the Sine Values:
To perform the calculations, we need the sine values of the angles:
- [tex]\(\sin(21°)\)[/tex]
- [tex]\(\sin(138°)\)[/tex]
Note: [tex]\(\sin(138°) = \sin(180° - 42°) = \sin(42°)\)[/tex], because sin(180° - x) = sin(x).
5. Compute the Other Sides:
[tex]\[
b = \frac{57 \cdot \sin(21°)}{\sin(138°)}
\][/tex]
Simplifying further,
[tex]\[
b = \frac{57 \cdot \sin(21°)}{\sin(42°)}
\][/tex]
6. Evaluate the Expression:
Given the numerical result,
[tex]\[
b = 30.5 \, \text{cm}
\][/tex]
So, the lengths of the other sides of the isosceles triangle are:
C. 30.5 cm
Thus, the correct answer is: C. 30.5 cm.
