A geometry problem by mario dawi

Geometry Level 2

A large helium balloon is tethered to the ground by two taut lines. One line is 100 feet long and makes an 80 ° angle with the ground. The second line makes a 40 ° angle with the ground. How long is the second line (x), to the nearest foot? How far apart are the tethers (y)?

y=95 x=105 y=205 x=150 y=135 x=153 y=125 x=142

Mario Dawi by mario dawi , 22, Canada

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2 solutions

By Sine Rule , we have

x sin ⁡ 80 = 100 sin ⁡ 40 \dfrac{x}{\sin 80}=\dfrac{100}{\sin 40}

x = sin ⁡ 80 sin ⁡ 40 × 100 ≈ 153 x=\dfrac{\sin 80}{\sin 40} \times 100 \approx \boxed{153}

By Sine Rule again, we have

y sin ⁡ 60 = 100 sin ⁡ 40 \dfrac{y}{\sin 60}=\dfrac{100}{\sin 40}

y = sin ⁡ 60 sin ⁡ 40 × 100 ≈ 135 y=\dfrac{\sin 60}{\sin 40} \times 100 \approx \boxed{135}

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By law of sines, we have

x sin ⁡ 80 = 100 sin ⁡ 40 \dfrac{x}{\sin~80}=\dfrac{100}{\sin~40} ⟹ \color{#D61F06}\large{\implies} x = 153 \color{plum}\boxed{x=153}

y sin ⁡ 60 = 100 sin ⁡ 40 \dfrac{y}{\sin~60}=\dfrac{100}{\sin~40} ⟹ \color{#D61F06}\large{\implies} y = 135 \color{plum}\boxed{y=135}

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