To find the measure of the smallest angle [tex]\(\angle Q\)[/tex] in a triangle with given side lengths 4.5 and 6, we can use the Law of Cosines.
The Law of Cosines states:
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc}, \][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the lengths of the sides of the triangle, and [tex]\(A\)[/tex] is the angle opposite the side [tex]\(a\)[/tex].
Given:
[tex]\(a = 4.5\)[/tex],
[tex]\(b = 6\)[/tex],
[tex]\(c = 6\)[/tex].
We want to find [tex]\(\angle A\)[/tex] because it is the angle opposite the smallest side, which is 4.5.
1. Compute [tex]\( \cos(A) \)[/tex]:
[tex]\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \][/tex]
Substitute the given values into the formula:
[tex]\[ \cos(A) = \frac{6^2 + 6^2 - 4.5^2}{2 \cdot 6 \cdot 6} \][/tex]
2. Calculate the values of the variables:
[tex]\[ 6^2 = 36 \][/tex]
[tex]\[ 4.5^2 = 20.25 \][/tex]
[tex]\[ 6 \cdot 6 = 36 \][/tex]
3. Plug in these values:
[tex]\[ \cos(A) = \frac{36 + 36 - 20.25}{36 \cdot 2} \][/tex]
[tex]\[ \cos(A) = \frac{71.75}{72} \][/tex]
4. Simplify:
[tex]\[ \cos(A) \approx 0.9965 \][/tex]
5. To find [tex]\(\angle A\)[/tex], take the arccos of 0.9965:
[tex]\[ A \approx \arccos(0.9965) \][/tex]
Using a calculator or a mathematical table for arccos:
[tex]\[ A \approx 44^\circ \][/tex]
Therefore, the smallest angle [tex]\( \angle Q \)[/tex] is roughly [tex]\( 44^\circ \)[/tex].
The closest option to this value is:
[tex]\[ 44^\circ \][/tex]
This confirms the measure of the smallest angle in the triangle is [tex]\( 44^\circ \)[/tex], rounded to the nearest whole degree.
