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Answer :
To find the measure of the angle with the least measure in a triangle with sides measuring 14.9 centimeters, 23.8 centimeters, and 36.9 centimeters, we can use the cosine rule. The cosine rule helps us find the angles of a triangle when we know the lengths of all three sides.
Here's a step-by-step guide:
1. Identify the sides of the triangle:
- [tex]\( a = 14.9 \)[/tex] cm
- [tex]\( b = 23.8 \)[/tex] cm
- [tex]\( c = 36.9 \)[/tex] cm
2. Use the cosine rule to find each angle:
- For angle [tex]\( A \)[/tex], which is opposite side [tex]\( a \)[/tex]:
[tex]\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\][/tex]
- For angle [tex]\( B \)[/tex], which is opposite side [tex]\( b \)[/tex]:
[tex]\[
\cos B = \frac{a^2 + c^2 - b^2}{2ac}
\][/tex]
- For angle [tex]\( C \)[/tex], which is opposite side [tex]\( c \)[/tex]:
[tex]\[
\cos C = \frac{a^2 + b^2 - c^2}{2ab}
\][/tex]
3. Convert the cosine values to degrees to find the angles:
- Angle [tex]\( A \)[/tex] is approximately 13.8°.
- Angle [tex]\( B \)[/tex] is approximately 22.3°.
- Angle [tex]\( C \)[/tex] is approximately 143.9°.
4. Identify the smallest angle:
- The smallest angle in the triangle is angle [tex]\( A \)[/tex], which measures about 13.8°.
Therefore, the angle with the least measure is approximately 13.8°, which corresponds to option d.
Here's a step-by-step guide:
1. Identify the sides of the triangle:
- [tex]\( a = 14.9 \)[/tex] cm
- [tex]\( b = 23.8 \)[/tex] cm
- [tex]\( c = 36.9 \)[/tex] cm
2. Use the cosine rule to find each angle:
- For angle [tex]\( A \)[/tex], which is opposite side [tex]\( a \)[/tex]:
[tex]\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\][/tex]
- For angle [tex]\( B \)[/tex], which is opposite side [tex]\( b \)[/tex]:
[tex]\[
\cos B = \frac{a^2 + c^2 - b^2}{2ac}
\][/tex]
- For angle [tex]\( C \)[/tex], which is opposite side [tex]\( c \)[/tex]:
[tex]\[
\cos C = \frac{a^2 + b^2 - c^2}{2ab}
\][/tex]
3. Convert the cosine values to degrees to find the angles:
- Angle [tex]\( A \)[/tex] is approximately 13.8°.
- Angle [tex]\( B \)[/tex] is approximately 22.3°.
- Angle [tex]\( C \)[/tex] is approximately 143.9°.
4. Identify the smallest angle:
- The smallest angle in the triangle is angle [tex]\( A \)[/tex], which measures about 13.8°.
Therefore, the angle with the least measure is approximately 13.8°, which corresponds to option d.
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