Geometry of triangles and circles

Question

A balloon is tethered to a peg in the ground by a 20m string which makes an angle of 72 degrees to the horizontal. An observer notes that the angle of elevation from him to the balloon is 41 degrees and his angle of depression to the peg is 10 degrees. Find the horizontal distance of the observer from the peg.

Answer 1

Hi Polina.

 

This is very hard to explain without being able to draw the triangles...

 

We are going to have drawn 4 different triangles before we are through - so make your drawing really big so that you can fit everything and not have it confusing...

 

First triangle:

 

Draw the right triangle with the balloon string as the hypotenuse, making a 72 degree angle with a horizontal line.  The "peg in the ground" should be the lower, left vertex, on the horizontal line.  The balloon should be the upper right vertex, somewhere above and to the right of the peg.  And the last side of this right triangle will be a vertical line from the balloon (upper right vertex) down to the ground (horizontal line.)

 

Now, let's move to the observer, the second triangle.

 

He is standing on the other side of the peg, to the left, along the horizontal line.  This distance, call "x" because it is the thing we are looking for.  The hypotenuse of this triangle is the line from the observer's eye down to the peg.  So this second triangle has a 90° angle as its bottom left vertex (made by the ground and the observer), its bottom right angle is at the peg, and the hypotenuse is the line you just drew from his eye to the peg.  Leave this for a moment, we will return to it at the very end.

 

Make a stick figure for the observer, and draw a horizontal line, very lightly, from the observers eye-level to the right.  This line should be barely visible and parallel to the ground.

         /

        / 41°                               O ^(balloon)

    O\-------------------------       /

   \|/ \  10°                            / 20 m

    |    \                                /

   / \                                  /72°

---|----------^X----------------|---------------

                                     ^(peg)

 

Draw a line from his eye to the balloon.  This should be 41° up from the eye-level horizontal line.  (Sorry, mine are angle wrong, no choices other than the "/" and "\" keys)

Draw a line from his eye down to the peg.  This should be 10° down from the eye-level horizontal line.  (Sorry, mine are angle wrong, no choices other than the "/" and "\" keys)

 

 

When you are finished with the drawing you should have the third triangle with a vertex at his eye, with a measure of 51°.  (Got this by adding the 41 from on top and the 10 from on the bottom).  Another vertex at the peg whose angle we will later determine to be 98°.  And the the third vertex at the balloon.  The side from the peg to the balloon should be 20 m  (it is still the balloon string).  This is NOT a right triangle.

 

Now we need that upper right angle of this third triangle, the one at the balloon.   If you extend the eye-level horizontal line to touch the vertical line of the first triangle (the right triangle on the right), you will get another right triangle from the observer's eye over to the vertical line.  This is only useful to find the angle of the vertex in the 3rd triangle.    The first triangle we drew on the right has 72° on the bottom left vertex, 90° on the bottom right vertex, and up at the balloon it should have (180 - 72 = 18)  18°.

 

Now look at the right triangle whose left vertex is the observer's eye.  This right triangle has its 90° angle all the way to the right, and its upper left angle is 18° + whatever the angle between the kite string and the new right triangle's hypotenuse.  This is how we will find the angle we are looking for.  The right triangle whose left vertex is at his eye has angles 41 + 90 + 18 + ? = 180.  It's measure must be 31°.

 

Finally, we have everything we need for our 2nd triangle, the NOT right triangle.

 

At his eye is a vertex with 51°, at the balloon is a vertex with 31°, and at the peg is a vertex of 180 - 51 - 31 = 98°.

The only side length we know is the kite string which is 20 m between the 31° angle and the 98° angle.

 

We can now use the Law of Sines to find the length of the side between the 51° angle and the 98° angle.

 

[sin(51)]/20 =  [sin(31)]/?      Solving for the "?" we get [(20) sin(31)]/[sin(51)]  and the length of the side between the 51° angle and the 98° angle is  13.2546 m

 

Now, to answer their question.

 

We want the distance from the observer to the peg.  If the observer is the vertical side of our 2nd triangle, this one with the vertical as the observer, the horizontal leg is x, the distance we are looking for, and the hypotenuse is the line from the observer's eye to the peg.

 

The length of this hypotenuse is the 13.2546 we just calculated.  The angle between this hypotenuse and the ground is 10°, and we want to know the length of the side adjacent to the 10° angle.  That calls for a cosine.

 

cos(10) = x/13.2546

 

x = 13.2546 cos(10) = 13.05 m

 

 

THAT was a hard problem!!!