Two points A and C are on the same level ground as the foot of pole B . The distance between A and C is 40m and A and C are on the same sides of the vertical pole . The distances from the top of the pole D to A and C are 53 and 85 respectively. Find correct to 1 do the distance between the foot of the pole B and the point A.

First, we have to illustrate the problem. I can't draw as of the moment but I'll try my best to describe the drawing as best as I can. You can draw this in a scratch paper to have a better understanding.

In a horizontal line, there is a point A, B, and C.
From left going to right, the arrangement of points will be C-A-B.
The distance between A and C is 40m.
The pole is standing at point B with height h.
The tip of the pole or the top most part of the pole is point D.
From point D to A, the measurement is 53m.
From point D to C, the measurement is 85m.
Let's name the distance between A and B as x.

Now that you have drawn this. There should be two triangles formed now namely triangle DCB and DAB.

TRIANGLE DCB

By Pythagorean Theorem, we can write an equation for the height of the pole.

[tex]h^{2} = {53}^{2} - {x}^{2} [/tex]

TRIANGLE DAB

By Pythagorean Theorem, we can write an equation for the height of the pole.

[tex] {h}^{2} = {85}^{2} - {(40 + x)}^{2} [/tex]

Since we only have one pole, this means that we can equate the two equations.

[tex] {53}^{2} - {x}^{2} = {85}^{2} - {(40 + x)}^{2} \\ {53}^{2} - {85}^{2} = {x}^{2} - ( {x}^{2} + 80x + 1600) \\ - 4416 = - 80x - 1600 \\ 80x = 4416 - 1600 \\ 80x = 2816 \\ x = \frac{2816}{80} \\ x = 35.2[/tex]