Question

Triangle X Y Z is shown. Angle X Y Z is 51 degrees and angle Y Z X is 76 degrees. The length of X Z is 2.6, the length of X Y is z, and Y Z is x.

Law of sines: StartFraction sine (uppercase A) Over a EndFraction = StartFraction sine (uppercase B) Over b EndFraction = StartFraction sine (uppercase C) Over c EndFraction

What is the value of z, rounded to the nearest tenth? Use the law of sines to find the answer.

2.7 units
3.2 units
4.5 units
5.3 units

Answer 1

Using the Law of Sines, we can find the third angle of the triangle, XYZ, which is 53 degrees.

Then, apply the Law of Sines to solve for the length of side XY, denoted as z, and find that z is approximately 2.7 units when rounded to the nearest tenth.

Step-by-step explanation:
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides of the triangle. In triangle XYZ, we know angle YZX = 51 degrees, angle ZYX = 76 degrees, and the length of side ZX = 2.6. To find the third angle, XYZ, we deduct the sum of the two known angles from 180 degrees because the total of internal angles in a triangle adds to 180 degrees which yields XYZ = 53 degrees.

Applying the Law of Sines, we can solve for the length of side XY or z as follows :

sin(76)/2.6 = sin(53)/z

This yields z = (2.6* sin(53))/ sin(76)

By calculating this expression, we find that z approximately equals 2.7 units when rounded to the nearest tenth.