Answer:
- long side: 19.1 cm
- missing angle: 25°
- hypotenuse: 21.1 cm
Explanation:
You can use this information to "guess" at the answers without doing any "work."
- The sum of angles in a triangle is 180°.
- The shortest side is opposite the smallest angle.
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qualitative solution
The missing angle is the complement of the marked acute angle in the right triangle, so is ...
C = 90° -65° = 25°
This angle is opposite the side of length 8.9 cm. The next-smallest angle is 65°, which is more than double the smallest angle. Hence the side opposite 65° will not be either of 3.8 or 9.8.
Of the two remaining measures, the longer one, 21.1, will be the hypotenuse, BC. The shorter of those, 19.1, will be the long side, AC.
Our solution is ...
- AC = 19.1 cm
- C = 25°
- BC = 21.1 cm
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quantitative solution
The mnemonic SOH CAH TOA reminds you of the relations between trig functions and right triangle sides. Here, we're given an angle and the length of its adjacent side. We are asked for the opposite side and for the hypotenuse. This suggests useful relations might be ...
Cos = Adjacent/Hypotenuse
Tan = Opposite/Adjacent
Solving the first of these for the hypotenuse gives ...
hypotenuse = adjacent/cos(65°)
BC = 8.9 cm/cos(65°) ≈ 21.059 cm ≈ 21.1 cm
Solving the second relation above for the opposite side gives ...
opposite = adjacent×tan(65°)
AC = 8.9 cm×tan(65°) ≈ 19.086 cm ≈ 19.1 cm
As above, the missing angle is the complement of the given one:
C = 90° -65° = 25°
Then the quantitative solution is ...
- AC = 19.1 cm
- C = 25°
- BC = 21.1 cm
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Additional comment
If AC were 9.8 cm, the angle at B would be about 48°. That is, the two acute angles in the triangle would be very nearly equal.
We know that the side ratios in a 30°-60°-90° right triangle are 1 : √3 : 2. This triangle has a larger angle greater than 60°, so its longer side will be more than √3 times the short side. That means a length of 9.8 cm is way too short.