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SPI Mathematics · Heights and Distances

Kites, Ropes and Slopes

When a kite string, a rope, a slide or a ramp is the slanting side of the triangle, it is the hypotenuse, and sin or cos joins it to the height or the ground distance.

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Why this matters

Eleven questions in the bank, seven EASY and four MODERATE. In the EASY ones a string, rope, slide or ramp is the slanting side, so the ratio is sin or cos instead of tan. The four MODERATE ones hide the triangle: a wire between two poles, a broken tree, or a line drawn inside a rectangle.

Concept 1 of 2: The slanting side is the hypotenuse

A tight string from a kite to the ground, a rope from a pole top to the ground, a slide or a ramp: each runs along the slanting side of a right triangle. That side is the hypotenuse, so the height opposite the angle uses sin and the ground distance next to it uses cos.

Definition

  • Height == slant length ×sin⁡θ\times \sin\theta, so slant length == height ÷sin⁡θ\div \sin\theta.
  • Ground distance == slant length ×cos⁡θ\times \cos\theta.
  • θ\theta is the angle the slant side makes with the ground.
  • 'No slack' or 'tightly stretched' means the string is a straight line, so the triangle is exact.

Height and ground distance from a slant length

h=Lsin⁡θ,d=Lcos⁡θ,L=hsin⁡θh = L\sin\theta, \qquad d = L\cos\theta, \qquad L = \frac{h}{\sin\theta}
  • Lthe slant length (string, rope, slide, ramp)
  • hthe height of its top end
  • dthe ground distance it covers

Worked example

A kite is flying at a height of 45 m and its string makes an angle of 60∘60^\circ with the ground. Find the length of the string.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Heights and Distances · Kites, Ropes and Slopes

A contractor planned to install a slide for the children to play in a park. If he prefers to have a slide whose top is at a height of 1.5 m and is inclined at an angle of 30∘30^\circ to the ground, then the length of the slide would be

Using tan with a slant length

tan joins the height and the ground distance only. When the string or rope length is given or asked, the hypotenuse is involved, so use sin or cos.

Swapping sin and cos

The angle sits at the ground. The height is across from it (sin) and the ground distance touches it (cos). A 30 m ladder at 30∘30^\circ reaches 15 m up the wall, and its foot is 15315\sqrt{3} m out.
Drill 6 more on the slanting side is the hypotenuse

Concept 2 of 2: Finding the triangle in a wire, a broken tree or a rectangle

Some questions draw no triangle at all. Make one: a wire between two pole tops rises only by the difference in their heights; a broken tree is a standing part plus a fallen part leaning to the ground; a line across a rectangle cuts off a right triangle at a corner.

Definition

  • Wire between two poles: draw a horizontal line from the top of the shorter pole. The upright side of the triangle is the difference in heights, and the wire is the hypotenuse.
  • Broken tree: the standing part is upright, the broken part is the hypotenuse, and the gap from the foot to where the top touches is the ground side. The full height is standing part plus broken part.
  • Rectangle: every corner is a right angle, so a line from one corner to a side makes a right triangle with two sides of the rectangle. Opposite sides are equal.

Worked example

The tops of two poles 18 m and 10 m high are joined by a wire that makes 30∘30^\circ with the horizontal. Find the length of the wire.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Heights and Distances · Kites, Ropes and Slopes

The top of a broken tree has its top end touching the ground at a distance 15 m from the bottom, the angle made by the broken end with the ground is 30∘30^\circ. Then length of broken part is

Using the taller pole's full height

A wire between poles of 20 m and 12 m rises only 8 m. The ground below the shorter pole's top is not part of the triangle.

Calling the broken part the whole tree

The broken part is only the hypotenuse. If the question asks for the tree's height before it broke, add the standing part to the broken part.
Drill 3 more on finding the triangle in a wire, a broken tree or a rectangle

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

  • The slanting side is the hypotenuse

    Height and ground distance from a slant length

    h=Lsin⁡θ,d=Lcos⁡θ,L=hsin⁡θh = L\sin\theta, \qquad d = L\cos\theta, \qquad L = \frac{h}{\sin\theta}

Watch out for (4)

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