Imagine a ball sitting on the other side of a stream. You want to know the distance to it, but you don’t You cannot cross or stretch a tape measure across the water. A triangle can help: instead of measure the entire path, we measure a safe section and observe two angles.

A triangle that reveals a measurement

Let’s call P the point we want to reach. On the margin, find a point A so that the segment from A to P forms a right angle with the margin. Mark another point B in the margin, the some distance from A.

At B, compare the direction back to A with the direction to P. When these directions form a angle $45°$, triangle ABP has:

  • a right angle at A;
  • an angle of $45°$ at B;
  • an angle of $45°$ in P, because the three angles of a triangle add up to $180°$.

The sides opposite the two angles of $45°$ have the same length. Therefore, $AB=AP$. From a distance from A to point P is equal to the stretch of margin that we can measure.

Example with steps

Suppose you marked B at 8 m from A. When looking from B to P, you saw that the angle with the direction BA is from $45°$. Then:

$$AP=AB=8\text{ m}.$$

The result does not say that the distance from B to P is also 8 m: that side is the hypotenuse and measures more. Identifying which distance is being calculated is as important as doing the math.

In everyday life

  • Our eyes. Each eye sees the object from a slightly different angle; the brain “solves the triangle” and estimates the distance. Closing one eye makes this more difficult.
  • Cell phones with two cameras. “Portrait mode” background blur uses the same idea: two lenses set apart compare angles and calculate the distance to each point.
  • Stars. Astronomers measure the distance to nearby stars by the difference in angle observed on opposite sides of Earth’s orbit (the parallax).
The two eyes form the base of a triangle: the further away the object, the smaller the angle between the views.

Try it with a nearby target

Ask an adult to choose a tree or other fixed target on safe, level ground. Don’t do the activity near streets, ravines or water.

  1. Using a paper square, find point A on the floor line where the direction to the target is perpendicular to the imaginary margin line.
  2. Walk along this line until you find a point B where the direction to the target forms $45°$ with BA. A card folded diagonally can serve as a guide for the angle.
  3. Measure AB with a tape or count steps of known size.
  4. Use $AP=AB$ to estimate the perpendicular distance to the target.

Small errors in marking the angles change the estimate. Therefore, repeat the observation and compare the results; Do not treat a measurement taken by eye as if it were exact.

Challenge

You measured 6 m from A to B, and the angle at B is $45°$. What is the perpendicular distance from A to P? Which side of the triangle did you not discover with this equality?

Check the answer

$AP=AB=6$ m. The equality does not directly give the length of the BP hypotenuse; to find you, it would be necessary to use another relationship.

This idea of locating a point using a measured line and observed directions is called triangulation. It appears on maps, land surveys and in many other situations where It is not possible to directly reach the place we want to measure.

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