Answer:
- Angle YXZ = 41.6°, Area = 16.6 cm²
- Angle YXZ = 138.4°, Area = 3.2 cm²
Explanation:
You want the measure of angle YXZ and the area of ∆XYZ, given XZ = 5.2 cm, YZ = 6.7 cm, and ∠Y = 31°.
Law of sines
The law of sines tells you triangle sides are proportional to the sine of the opposite angle. The given angle is opposite the shorter side, so there will be two (2) solutions to this triangle. One solution for angle X is the supplement of the other solution for angle X.
sin(X)/x = sin(Y)/y . . . . . . . law of sines
sin(X) = (x/y)sin(Y) . . . . . . solving for angle X
X = arcsin(6.7/5.2·sin(31°)) ≈ 41.6° or 138.4°
Angle YXZ can be either 41.6° or 138.4°.
Angle sum
The Angle Sum theorem tells you the sum of angles in a triangle is 180°, so the measure(s) of angle Z will be ...
Z = 180° -Y -X
Z = 180° -31° -{41.6, 138.4}° = {107.4°, 10.6°}
Area
The area of the triangle can be found using the formula ...
A = (1/2)xy·sin(Z)
Using the given values for x = YZ and y = XZ, we have ...
A = (1/2)(6.7 cm)(5.2 cm)·sin({107.4°, 10.6°} = {16.6, 3.2} cm²
For ∠X = 41.6°, the area is 16.6 cm²; for ∠X = 138.4°, the area is 3.2 cm².
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Additional comment
The first attachment shows the calculator results for angles and area, with full precision and rounded to 1 decimal place. The other two attachments show the results from a triangle solver using the given sides and angle.
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