Surveying......-to measure the angle DAE with a tape , each of the distances AB and AC was made 60 ft.?

Question

If the distance from B to C was then found to be 89.5 ft, the angle DAE was......?

Answer 1

I don't know where D and E are so I can't speak to the angle DAE, but I can help with the three angles of the triangle ABC.

In the general case, when the three sides - a, b, c - are all different, you can start with Heron's formula to get the area:

let s = (a + b + c)/2
then the area S = sqrt(s(s-a)(s-b)(s-c))

but S is also equal to (1/2)ab sin(included angle)

So you can solve for the sine of each included angle and then look up the arc-sine.

In this case, you seem to have a special case so you can take a shortcut.

Since AB = AC, you have an isosceles triangle. Drop the perpendicular from A to the side BC. It will land at the middle of the side at P.

Then APB = APC = right angle so
(BP)/(BA) = cos(PBA)

BP is half of BC, and you know BA so you can compute the cosine without square roots. Then look up the arc cosine and you have two of the three angles. Since the sum of the three is 180 degrees, and you know two, you can compute the third.