samphacc

2022-09-27

Triangle J K L is shown. The length of J K is 13, the length of K L is 11, and the length of L J is 19.Law of cosines: a2 = b2 + c2 – 2bccos(A)

Find the measure of AngleJ, the smallest angle in a triangle with sides measuring 11, 13, and 19. Round to the nearest whole degree.

30°
34°
42°
47°

nick1337

To find the measure of angle J, the smallest angle in triangle JKL with side lengths 11, 13, and 19, we can use the Law of Cosines. The Law of Cosines states that for a triangle with sides of lengths a, b, and c, and opposite angles A, B, and C, respectively, the following equation holds:
${a}^{2}={b}^{2}+{c}^{2}-2bc\mathrm{cos}\left(A\right)$
In this case, we want to find the measure of angle J, which is opposite side length 11 (let's call it side a). The other two side lengths are 13 (let's call it side b) and 19 (let's call it side c).
Using the Law of Cosines, we can rearrange the equation to solve for angle J:
$\mathrm{cos}\left(J\right)=\frac{{b}^{2}+{c}^{2}-{a}^{2}}{2bc}$
Plugging in the values, we have:
$\mathrm{cos}\left(J\right)=\frac{{13}^{2}+{19}^{2}-{11}^{2}}{2·13·19}$
Calculating this expression, we find:
$\mathrm{cos}\left(J\right)=\frac{169+361-121}{494}=\frac{409}{494}$
Now, we can use the inverse cosine (or arccosine) function to find the measure of angle J:
$J={\mathrm{cos}}^{-1}\left(\frac{409}{494}\right)$
Using a calculator or mathematical software, we can determine the value of J, rounded to the nearest whole degree.
Based on the provided options, the measure of angle J is approximately 47°.

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