Answer:
- YZ ≈ 8.5
- ∠X ≈ 70.5°
- ∠Z ≈ 19.5°
Explanation:
You want the solution to a right triangle with a side length of 3 and a hypotenuse of 9.
YZ
The length of the missing side can be found using the Pythagorean theorem:
XZ² = XY² +YZ²
YZ² = XZ² -XY²
YZ = √(XZ² -XY²) = √(9² -3²) ≈ 8.5
The length of YZ is about 8.5 units.
Angle X
The hypotenuse and side adjacent angle X are given, so we can use the cosine relation:
Cos = Adjacent/Hypotenuse
cos(X) = 3/9
X = arccos(3/9) ≈ 70.5°
The measure of angle X is about 70.5°.
Angle Z
Angle Z can be found two ways:
- the complement of angle X
- using the inverse sine function
The sine relation is ...
Sin = Opposite/Hypotenuse
sin(Z) = 3/9
Z = arcsin(3/9) ≈ 19.5°
Or ...
Z = 90° -X = 90° -70.5° = 19.5°
The measure of angle Z is about 19.5°.
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Additional comment
It is generally easier to find the measure of the second angle as the complement of the first. In the attached calculation, we wanted to show all of the answers on one line, so we didn't want to depend on previous results in order to find the remaining angle. For this purpose, using the trig solution was preferred. (Note that the calculator display has the results in the order {YZ, ∠Z, ∠X}.)
Of course the fraction 3/9 can be reduced to 1/3. There is no need.