Question

ind the perimeter of triangle, G, H, I△GHI. Round your answer to the nearest tenth if necessary. Figures are not necessarily drawn to scale.
D
E
F
G
H
I
31°
45°
31°
104°
19
10
16
x
14
22.4

Answer 1

I can solve it, but I need one clarification: how are the given numbers and angles attached to the diagram? In particular, which two side lengths and which angle belong to triangle GHI? (For example: is GH = 14, HI = 22.4, and ∠GHI = 104°?)

If that is the configuration, use the Law of Cosines to find the third side GI and then add the three sides for the perimeter. Calculation:

- GI^2 = GH^2 + HI^2 − 2(GH)(HI) cos ∠GHI
- GI^2 = 14^2 + 22.4^2 − 2(14)(22.4) cos 104°
- GI^2 ≈ 196 + 501.76 − 627.2(cos 104°)
(cos 104° ≈ −0.2419219)
- GI^2 ≈ 697.76 − 627.2(−0.2419219) ≈ 697.76 + 151.79 ≈ 849.55
- GI ≈ √849.55 ≈ 29.15

Perimeter ≈ 14 + 22.4 + 29.15 = 65.55 → 65.6 (to the nearest tenth).

If my assignment of labels is incorrect, please upload the figure or tell me which segments/angles correspond to G, H, and I and I’ll compute the correct perimeter.