KJSEA 2025 Maths Q36 — Angle of Depression (Height of a Building)

KJSEA 2025 Grade 9 Trigonometry

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The Question

“Dowdy stood on a balcony and saw a ball on the ground 47 m from the foot of the building. The angle of depression of the ball was 35 degrees. Calculate the height from the ground up to Dowdy.”

1

Draw and label the right-angled triangle

Sketch the building as a vertical line with Dowdy at the top and the ball on the ground. The ground, the wall, and Dowdy's line of sight to the ball form a right-angled triangle. The right angle is at the foot of the building, the height we want is the vertical side, and the 47 m is the horizontal distance along the ground.

The foot of the building gives the 90 degree corner because the wall is vertical and the ground is horizontal.

2

Move the angle into the triangle

The angle of depression is measured at the top, from the horizontal looking out down to the line of sight. Because the horizontal at the top and the ground are parallel lines, the angle of depression equals the angle at the ball looking back up to Dowdy. These are alternate angles, so the angle inside the triangle at the ball is also 35 degrees.

angle of depression=angle of elevation at the ball=35∘\text{angle of depression} = \text{angle of elevation at the ball} = 35^\circ
3

Choose the correct trigonometric ratio

Sitting at the 35 degree angle at the ball, the height is the side opposite the angle and the 47 m is the side adjacent to it. The ratio that links the opposite and adjacent sides is the tangent, so tangent of the angle equals the height divided by 47.

tan⁡35∘=h47\tan 35^\circ = \frac{h}{47}
4

Solve for the height

Multiply both sides by 47 to make the height the subject, then put in the value of the tangent of 35 degrees, which is about 0.70, and work out the product.

h=47×tan⁡35∘h = 47 \times \tan 35^\circ
h=47×0.70≈32.9 mh = 47 \times 0.70 \approx 32.9 \text{ m}

Final Result

The height from the ground up to Dowdy is about 32.9 metres.

Why this method works

The method works because the person, the base of the building, and the ball form a right-angled triangle in which we know one acute angle and the side next to it. Trigonometric ratios connect an angle to a pair of sides, and tangent is exactly the ratio of the opposite side to the adjacent side. The angle of depression is not inside the triangle, but the horizontal line at the top is parallel to the ground, so alternate angles let us copy that 35 degrees to the ball, where it becomes a usable angle. Knowing the angle and the adjacent side then fixes the opposite side completely.

✓

Check: 32.9 divided by 47 gives about 0.70, which is the tangent of 35 degrees, so the height is consistent with the given angle.