Question

1. use the tangent ratio and the pythagorean theorem to find perimeter and the area of this triangle, both to the nearest tenth of a unit. (diagram: right triangle ( ghf ) with right angle at ( g ), ( angle h = 42^circ ), ( gh = 2.7 , \text{cm} ))

Explanation:

Step1: Find length of FG

In right triangle \( \triangle FGH \), \( \angle H = 42^\circ \), \( GH = 2.7 \) cm.
\( \tan(42^\circ) = \frac{FG}{GH} \)
\( FG = GH \cdot \tan(42^\circ) \approx 2.7 \cdot 0.9004 \approx 2.431 \) cm

Step2: Find length of FH

Using Pythagorean Theorem:
\( FH = \sqrt{FG^2 + GH^2} \approx \sqrt{(2.431)^2 + (2.7)^2} \approx \sqrt{5.909 + 7.29} \approx \sqrt{13.199} \approx 3.633 \) cm

Step3: Calculate Perimeter

Perimeter \( = FG + GH + FH \approx 2.431 + 2.7 + 3.633 \approx 8.764 \approx 8.8 \) cm

Step4: Calculate Area

Area \( = \frac{1}{2} \cdot FG \cdot GH \approx \frac{1}{2} \cdot 2.431 \cdot 2.7 \approx 3.282 \approx 3.3 \) \( \text{cm}^2 \)

Answer:

Perimeter: \( \approx 8.8 \) cm, Area: \( \approx 3.3 \) \( \text{cm}^2 \)