Question

The figure below shows a flying kite. At a certain moment, the kite string forms an angle of elevation of 75° from point A on the ground. At the same moment, the angle of elevation of the kite at point B, 240 ft from A on level ground, is 45°. What is the length, in feet, of the string?

Answer 1

  To get the measures of the angles and the sides of a triangle we use sine rule,

Option B will be the correct option.

From the triangle given in the figure,

• Angle of elevation of a kite which is at point C from point B is 45°.
• Angle formed by the string and the horizontal at point A is 45°

     Let the length of the string of the kite = l feet                    

     By applying triangle sum theorem in the given triangle ABC,

m(∠A) + m(∠B) + m(∠C) = 180°

75° + 45° + m(∠C) = 180°

m(∠C) = 180° - 120°

m(∠C) = 60°

      Now apply the sine rule in the given triangle ΔABC,

$\frac{\text{sin}(B)}{AC}=\frac{\text{sin}(C)}{AB}$

$\frac{\text{sin}(45^{\circ})}{l}=\frac{\text{sin}(60^{\circ})}{240}$

$l=\frac{\text{sin}(45^{\circ})\times 240}{\text{sin}(60^{\circ})}$

$l=\frac{\frac{1}{\sqrt{2}}\times 240}{\frac{\sqrt{3} }{2} }$

$l=\frac{240}{\sqrt{2}}\times \frac{2}{\sqrt{3} }$

$l=\frac{480}{\sqrt{6}}$

$l=\frac{480}{\sqrt{6} }\times \frac{\sqrt{6} }{\sqrt{6} }$

$l=\frac{480\sqrt{6} }{6}$

$l=80\sqrt{6}$

Therefore, Option (B) will be the answer.

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Answer 2

Here we want to use what we know about trigonometry to find the side of a triangle.

We will see that the correct option is B.

First, we know that the sum of all internal angles of a triangle is 180°.

Then the missing angle of the triangle, let's call it C, is given by:

A + B + C = 180°

75° + 45° + C = 180°

C = 180° - 45° - 75° = 60°.

Now we can use the sine rule, it says that the sine of an angle over the length of the opposite side of the triangle is a constant, then we have:

$\frac{sin(A)}{BC} = \frac{sin(B)}{AC} = \frac{sin(C)}{AB}$

Here the length of the string is AC.

And we know that:

B = 45°

C = 60°

AB = 240

Then we can use the second and third parts of the above equality to get:

$\frac{sin(45 \°)}{AC} = \frac{sin(60 \°)}{240}$

Now we can solve this for AC:

$AC = \frac{sin(45 \°)}{sin(60 \°)}*240 = \frac{2 }{\sqrt{3*2}}*240$

Now we can multiply and divide by √6

$AC = \frac{\sqrt{6} }{\sqrt{6} } *\frac{2}{\sqrt{6} }*240 = \frac{\sqrt{6} }{3}*240 = \sqrt{6} *80$

Then we can see that the correct option is B.

If you want to learn more, you can read:

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