Question

Show how to calculate the distance between two inaccessible points \(A, B\), by the use of similar triangles. (Assume, for example, that the two points are on the bank of a river opposite your position.)

Short answer

Answer: To find the distance between two inaccessible points A and B, we can follow these steps: 1. Choose a reference point (C) on the same side of the river and measure the distances between points C, A, and B (denoted as \(d_{AC}\) and \(d_{BC}\)). 2. Measure the angles \(∠ACB\) and \(∠CAB\), denoted as \(\alpha\) and \(\beta\), using an angle measuring instrument. 3. Create two similar triangles (\(\triangle ADC\) and \(\triangle BCD\)) by drawing a line segment CD parallel to AB. 4. Use the properties of similar triangles to set up a proportion: \(\frac{d_{AD}}{d_{AC}} = \frac{d_{AB}}{d_{AD}+d_{BC}}\). 5. Rearrange the equation to solve for the distance between points A and B: \(d_{AB} = \frac{d_{AD}}{d_{AC}} * (d_{AD} + d_{BC})\). 6. Substitute the measured values and the calculated value of \(d_{AD}\) into the equation to find the distance between points A and B.

Step-by-step solution

Choose a reference point and measure distances

Find a suitable reference point (C) on the side of the river where you are standing. Measure the distance between points C, A, and B (denoted as \(d_{AC}\) and \(d_{BC}\)) using a measuring tape or any distance-measuring tool.

Measure angles

Using a theodolite or any angle measuring instrument, measure the angles \(∠ACB\) and \(∠CAB\). We will denote these angles as \(\alpha\) and \(\beta\), respectively.

Create two similar triangles

Let point D be a point on the same side of the river as point A, such that the line segment CD is parallel to AB. This will make triangles \(\triangle ADC\) and \(\triangle BCD\) similar. Thus, we will have \(\angle ADC = \angle BCD = \alpha\) and \(\angle ACD = \angle CBD = \beta\).

Use the properties of similar triangles

Since triangles \(\triangle ADC\) and \(\triangle BCD\) are similar, we have the following proportion between their corresponding sides: \(\frac{d_{AD}}{d_{AC}} = \frac{d_{AB}}{d_{AD}+d_{BC}}\) We want to find the distance \(d_{AB}\) between points A and B.

Solve for the length of \(d_{AB}\)

Rearrange the equation from step 4 to solve for \(d_{AB}\): \(d_{AB} = \frac{d_{AD}}{d_{AC}} * (d_{AD} + d_{BC})\) Since we have already measured \(d_{AC}\) and \(d_{BC}\) in step 1, we can substitute these values into the equation and solve for \(d_{AB}\).

Calculation and result

Substitute the measured values \(d_{AC}\), and \(d_{BC}\), and the calculated value of \(d_{AD}\) into the equation for \(d_{AB}\) from step 5 and solve for the distance between the two inaccessible points A and B: \(d_{AB} = \frac{d_{AD}}{d_{AC}} * (d_{AD} + d_{BC})\) The calculated value of \(d_{AB}\) is the distance between the two inaccessible points A and B on the banks of the river.

Key concepts

Similar Triangles

In trigonometry, similar triangles are essential for understanding numerous geometric problems. Two triangles are considered similar if they have corresponding angles that are equal and their sides are in proportion. This property is extremely useful in solving various real-world problems, such as calculating the distance between two points that cannot be directly measured, like in this exercise with points A and B across a river.
To establish triangle similarity, it often involves creating auxiliary lines or points, like our reference point C and parallel segment CD in the solution. This method helps transform the problem into two similar triangles,

• \(\triangle ADC\) and \(\triangle BCD\)

with known measures and angles that allow us to calculate the desired distance using the properties of these triangles. Remember that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. This creates a powerful tool for solving problems indirectly and can be leveraged even in complex scenarios.

Angle Measurement

Measuring angles is a crucial aspect when dealing with trigonometric problems and can often dictate the accuracy of your calculations. In our exercise, measuring the angles \(\angle ACB\) and \(\angle CAB\), denoted as \(\alpha\) and \(\beta\) respectively, is necessary for setting up similar triangles.
Accurate angle measurement typically requires specific instruments, like a theodolite, which can measure horizontal and vertical angles with precision. For most practical applications, a clear line of sight and proper alignment of the instrument help in achieving reliable readings.
When measuring angles:

• Ensure the instrument is level and properly calibrated.
• Take multiple readings to average out any potential observational errors.

These measurements provide the basis for forming relationships between angles and distances that are further used to solve problems involving inaccessible distances, as in the given problem.

Geometric Proportions

Geometric proportions are pivotal when solving for unknown lengths using trigonometry, especially with similar triangles. The principle of proportions helps us set up an equation that relates the known and unknown lengths of sides in similar triangles. In our exercise, after establishing similar triangles, we derive an expression:
\[\frac{d_{AD}}{d_{AC}} = \frac{d_{AB}}{d_{AD}+d_{BC}}\]
This equation involves the proportions of the known lengths \(d_{AC}\) and \(d_{BC}\), and the unknown \(d_{AB}\), which we are solving for. To solve the equation, you follow algebraic manipulation to isolate \(d_{AB}\):
\[d_{AB} = \frac{d_{AD}}{d_{AC}} * (d_{AD} + d_{BC})\]
With the measured and calculated values, you can determine the desired distance effectively. By understanding these proportions, you can extend this technique to many similar trigonometric problems, where direct measurement is difficult but solving through geometric similarity is possible.