Looking at the top of tower A and base of tower B from points C and D, we find that ∠ACD = 60°, ∠ADC = 75° and ∠ADB = 30°. Let the distance between points C and D be 100. Find the height AB of the tower. Picture attached for the problem. Please show your work too. Thanks!

Step-by-step explanation:

We can use the Law of Sines to find segment AD, which happens to be a leg of and the hypotenuse of .

The Law of Sines states that the ratio of any angle of a triangle and its opposite side is maintained through the triangle:

Since we're given the length of CD, we want to find the measure of the angle opposite to CD, which is . The sum of the interior angles in a triangle is equal to 180 degrees. Thus, we have:

Now use this value in the Law of Sines to find AD:

Recall that and :

Now that we have the length of AD, we can find the length of AB. The right triangle is a 30-60-90 triangle. In all 30-60-90 triangles, the side lengths are in the ratio , where is the side opposite to the 30 degree angle and is the length of the hypotenuse.

Since AD is the hypotenuse, it must represent in this ratio and since AB is the side opposite to the 30 degree angle, it must represent in this ratio (Derive from basic trig for a right triangle and ).

Therefore, AB must be exactly half of AD:

Step-by-step explanation:

the triangle ∆ABD is a special right angle triangle of which we want to figure out length of its shorter leg (AB).

to do so we need to find the length of AD (the hypotenuse). With the help of ∆ADC it can be done. so recall law of sin

we'll ignore B/sinB as our work will be done using the first two

step-1: assign variables:

step-2: substitute

recall unit circle therefore:

simplify:

since ∆ABD is a 30-60-90 right angle triangle of which the hypotenuse is twice as much as the shorter leg thus:

simplify division:

and we're done!

$9.41

step-by-step explanation: