Heights and Distances with Two Triangles: Quick Revision

A brief note on Heights and Distances with Two Triangles. Read it before a test. Try each question at the end before you open its answer.

The idea

• One angle gives one equation, and there are two unknowns: the height and a distance.
• A second angle gives a second triangle and a second equation.
• A distance on the ground is the height times a cotangent: $x = h\cot\theta$.
• The nearer point sees the larger angle.

The four cases

• Two points, same side. The distances subtract: $h = \dfrac{d}{\cot\alpha - \cot\beta}$.
• Two points, opposite sides. The distances add: $h = \dfrac{d}{\cot\alpha + \cot\beta}$.
• A tower on a building. Both triangles share the base $x$, and the top part is $t = x(\tan\beta - \tan\alpha)$.
• A moving object. The height is fixed. The distance moved is $h\cot\beta - h\cot\alpha$, and the speed is the distance divided by the time.

Useful facts

• $\cot 30° = \sqrt{3}$, $\cot 45° = 1$ and $\cot 60° = \dfrac{1}{\sqrt{3}}$.
• An angle of depression equals the angle of elevation seen from the other end.
• When two angles add up to $90°$, $h^2$ equals the product of the two distances.
• A river's width is the difference of the distances to its two banks.

A worked example

A tower is seen from two points on one side of it, $80$ m apart. The angle of elevation is $30°$ at the farther point and $60°$ at the nearer one. Find the height.

1. Nearer triangle. $h = x\sqrt{3}$.
2. Farther triangle. $h = \dfrac{x + 80}{\sqrt{3}}$.
3. Equate. $3x = x + 80$, so $x = 40$ m.
4. The height. $h = 40\sqrt{3} \approx 69.28$ m.

Check with the shortcut: $\dfrac{80}{1.1547} = 69.28$ m.

Practice with new numbers

A program to try

The program finds the height from two points for three different cases, and checks the nearer distance.

Mistakes to avoid

• Taking $d$ as a distance to the foot. It is the gap between the two points.
• Adding on the same side, or subtracting on opposite sides.
• Giving the height of the top when the length of the pole is asked.
• Swapping the angles. The nearer point has the larger one.

Check yourself

1. Two points on one side of a tower are $24$ m apart. The angles of elevation are $30°$ and $60°$. How tall is the tower?

2. A flagstaff stands on a building. From a point $50$ m from the building, the angles of elevation of its bottom and top are $30°$ and $45°$. How long is the flagstaff?

3. An aeroplane flies level at $600$ m. Its angle of elevation is $45°$ and then $30°$. How far does it fly in between?

4. Two points on opposite sides of a tower are $100$ m apart. The angle of elevation of the top is $45°$ from each. How tall is the tower?