Answer:
x ≈ 50.1 or 49.0 — depending on how you solve the triangle
Explanation:
In the given figure, ∆IJK is marked with angles that make it similar to ∆LMN. Side lengths are given as j = 77, k = 60, m = x, n = 39, and you are asked to find the length of x.
Similar triangles
Angle J can be found from the other angles in ∆IJK.
J = 180° -I -K
J = 180° -63° -48° = 69°
You will observe that angles J and K have measures 69° and 48°, respectively, and that these are congruent to corresponding angles M and N. Since corresponding angles are congruent, the triangles are similar by the AA similarity postulate.
Corresponding sides
Using the given side lengths, we can write a proportion for corresponding sides, assuming similar triangles:
m/n = j/k
x/39 = 77/60 . . . . . . use the given values
x = 39·77/60 = 50.05 ≈ 50.1
The length of side x is about 50.1 units.
Solved — alternate solution
If we ignore triangle IJK and solve triangle LMN using the law of sines, we have ...
m/n = sin(M)/sin(N) . . . . . sides are in proportion to sin(opposite ∠)
x/39 = sin(69°)/sin(48°)
x = 39·sin(69°)/sin(48°) ≈ 49.0
Solving the triangle gives x = 49.0.
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Additional comment
Triangle IJK is "overspecified." The figure of triangle IJK given cannot be drawn to scale because the angles and side lengths are inconsistent. This puts us in the awkward position of having to guess the procedure you're intended to use to answer the question.
The first solution above is based on relations between similar triangles. The proportion of corresponding sides doesn't consider that those side lengths are inconsistent with the angles shown.
The second solution above is based only on the information given in ∆LMN. It uses the combination of given sides and angles, without reference to ∆IJK. We prefer this solution, but we suspect your grader is looking for the other solution.