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A scalene triangle has side lengths of 6, 11, and 12. Keyla uses the law of cosines to find the measure of the largest angle. Complete her work and find the measure of angle [tex]$Y$[/tex] to the nearest degree.

1. [tex]$12^2 = 11^2 + 6^2 - 2(11)(6) \cos(Y)$[/tex]
2. [tex]$144 = 121 + 36 - (132) \cos(Y)$[/tex]
3. [tex]$144 = 157 - (132) \cos(Y)$[/tex]
4. [tex]$-13 = -(132) \cos(Y)$[/tex]

[tex]Y \approx \square[/tex] degrees



Answer :

GinnyAnswer
Certainly! Let's help Keyla find the measure of angle \( Y \) step-by-step.

Given:
- Side lengths of the triangle: \( a = 11 \), \( b = 6 \), \( c = 12 \)
- Equation derived from the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(Y) \)
[tex]\[ 12^2 = 11^2 + 6^2 - 2(11)(6) \cos(Y) \][/tex]

1. Calculate \( 12^2 \):
[tex]\[ 12^2 = 144 \][/tex]

2. Calculate \( 11^2 + 6^2 \):
[tex]\[ 11^2 = 121 \][/tex]
[tex]\[ 6^2 = 36 \][/tex]
[tex]\[ 121 + 36 = 157 \][/tex]

3. Substitute these values back into the Law of Cosines equation:
[tex]\[ 144 = 157 - 2(11)(6) \cos(Y) \][/tex]

4. Calculate the product \( 2(11)(6) \):
[tex]\[ 2(11)(6) = 132 \][/tex]

5. Substitute the final calculations back into the equation:
[tex]\[ 144 = 157 - 132 \cos(Y) \][/tex]

6. Solve for \( \cos(Y) \):
[tex]\[ 144 = 157 - 132 \cos(Y) \][/tex]
[tex]\[ 144 - 157 = -132 \cos(Y) \][/tex]
[tex]\[ -13 = -132 \cos(Y) \][/tex]
[tex]\[ \cos(Y) = \frac{-13}{-132} \][/tex]
[tex]\[ \cos(Y) = \frac{13}{132} \][/tex]
[tex]\[ \cos(Y) \approx 0.09848484848484848 \][/tex]

7. Use the inverse cosine function to find \( Y \):
[tex]\[ Y = \cos^{-1}(0.09848484848484848) \][/tex]

8. Convert the resulting radians to degrees:
[tex]\[ Y \approx 1.4721515742803193 \text{ radians} \][/tex]
[tex]\[ Y \approx 84^\circ \][/tex]

So, the measure of angle [tex]\( Y \)[/tex] is approximately [tex]\( \boxed{84}^\circ \)[/tex].

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