Comprehension

From the top of a mobile tower, two steel wires are stretched to either side and attached to the ground, making angles 70° and 48°. The distance between the feet of the wires is 40 metres.

Question: 1

Draw a rough sketch showing these details.

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Drawing a clear and well-labelled diagram is the most critical first step in solving trigonometry word problems. It helps you identify the right-angled triangles and the relationships between their sides and angles.

Updated On: Mar 16, 2026

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Solution and Explanation

We need to create a diagram that visually represents the described scenario: a vertical tower with two support wires anchored to the ground on opposite sides.

1. A vertical line segment to represent the tower. Label the top T and the base B.
2. A horizontal line passing through B to represent the ground.
3. Two points, P and Q, on the horizontal line, located on opposite sides of B. These are the anchor points.
4. Line segments TP and TQ to represent the wires.
5. Label the angles the wires make with the ground: TPB = 70^ and TQB = 48^. 6. Label the total distance between the anchor points: PQ = 40 m.

The sketch will show two right-angled triangles, TBP and TBQ, which share a common side TB (the height of the tower). The bases of these triangles, BP and BQ, lie on the same straight line and add up to 40 metres.

The final answer is the diagram itself, correctly drawn and labelled with all the given information (tower, ground, wires, angles, and distance).

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Question: 2

What is the height of the tower ?

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This type of problem, with a central height and two angles on opposite sides, has a general solution form. If the total distance is d and angles are A and B, the height is h = (d)/( A + B). Memorizing this can be a shortcut.

Updated On: Mar 16, 2026

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Solution and Explanation

Using the setup from the sketch, we need to find a mathematical expression for the height of the tower, h.

We will use the tangent ratio, (θ) = OppositeAdjacent, in both right-angled triangles. Let the height be h = TB, and let the base segment BP = x. Then the other base segment is BQ = 40 - x.

In right-angled TBP:
(70^) = (TB)/(BP) = (h)/(x) x = (h)/((70^)) = h (70^) --- (1) In right-angled TBQ:
(48^) = (TB)/(BQ) = (h)/(40-x) 40-x = (h)/((48^)) = h (48^) --- (2) Substitute x from equation (1) into the expression 40-x:
BQ = 40 - h (70^) From equation (2), we know BQ = h (48^). Therefore:
h (48^) = 40 - h (70^) Now, solve for h. Move all terms with h to one side:
h (48^) + h (70^) = 40 Factor out h:
h ((48^) + (70^)) = 40 Isolate h:
h = (40)/((48^) + (70^)) This is the exact expression for the height. A numerical value would require a calculator.

The height of the tower is h = (40)/((70^) + (48^)) metres.

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