Foundations — Applications — heights and distances problems

This page assumes nothing. Before you can solve a single tower problem in Applications — heights and distances problems, you must be able to look at a picture and name every part without hesitation. So we build each symbol from the ground up, in the order they depend on each other.

1. A point and a line segment — the raw material

Why we need this: every problem starts by naming three points and joining them. If $AB$ sometimes means "the line" and sometimes "how long the line is", that is fine — context tells you which, and both readings live on the same picture.

Figure s01 shows two orange dots labelled $A$ and $B$ with a single teal segment drawn between them; the caption reminds you that the label $AB$ means both the segment and its length.

2. Angle and degrees — measuring "how much turn"

Why the topic needs this: the only thing a surveyor can actually measure while standing still is an angle — you point an instrument at the top of a tower and read off how far above horizontal you had to tilt. Everything else (heights, distances) is computed from that one measured angle.

Figure s02 draws a horizontal ink line and a teal line of sight tilting up from the same point; the orange arc between them is the angle $\theta$, measured in degrees from the horizontal.

3. The right angle — the anchor of everything

Now that we know what a degree is, we can name the most important angle of all.

The picture: where a vertical thing (a tower, a flagpole, a cliff face) meets the horizontal ground, they form a perfect corner. That corner is the right angle.

A triangle that contains a right angle is called a right triangle (or right-angled triangle). See 1.3.07-Pythagoras-theorem, which is a rule that works only inside such triangles.

4. Horizontal, vertical, and the line of sight

The picture: stand on flat ground and look up at a tower top. The horizontal goes out from your eye parallel to the ground. The line of sight tilts upward to the tower top. The vertical is the tower itself.

These three lines are the three sides of your right triangle:

• horizontal ground distance = one leg,
• vertical height = the other leg,
• line of sight = the slanted hypotenuse (defined in §6).

5. Angle of elevation and angle of depression

Why from the horizontal and not from the ground or the tower? Because the horizontal is the one direction that is the same no matter where you stand. It is the fair, fixed reference the instrument is levelled against.

Before the next box, three small ideas we will lean on:

Figure s03 shows a cliff-top point $O$ and a boat $P$, the two dashed parallel horizontals (with arrowheads), the teal line of sight acting as transversal, and the two equal orange arcs — depression at $O$, elevation at $P$ — labelled as alternate interior angles.

6. Naming the three sides relative to the angle

This is the single most important — and most misread — idea. The names of the sides depend on which angle you are looking from, not on where they physically sit.

Figure s04 draws the right triangle $A$–$C$–$B$ (right angle at $C$, marked with a square) with the angle $\theta$ at $A$: the teal ground leg is labelled adjacent, the plum vertical leg opposite, and the orange slant hypotenuse.

Because these labels come from the angle, you must always draw the triangle and mark the angle first, then read off which side is which.

7. The ratios — tan, sin, cos (why they exist)

Why they are trustworthy — shown right here. Look at Figure s05: it draws the same angle $\theta$ at the origin, with a small triangle and a big triangle sharing that angle. The small triangle has legs (opposite $=1$, adjacent $=2$); the big one is exactly double (opposite $=2$, adjacent $=4$). Compute $\tan \theta$ in each:

$\tan {\theta }_{\text{small}}=\frac{1}{2},\tan {\theta }_{\text{big}}=\frac{2}{4}=\frac{1}{2}.$

The size cancelled — same angle, same ratio. That is because both triangles are similar (same shape, different scale; see 3.2.04-Similar-triangles): doubling every side leaves every side-ratio unchanged. So each of $\sin \theta$, $\cos \theta$, $\tan \theta$ depends only on $\theta$, never on how big you drew the triangle. That constancy is the entire engine of the subject — full detail lives in 2.4.01-Trigonometric-ratios.

The symbol $⟹$ means "therefore / which gives". The symbol $\theta$ is our measured angle, $d$ the measured distance, $h$ the unknown we solve for.

8. Special angle values, and what happens at the edges

The symbol $\sqrt{3}$ ("square root of 3") is the number that multiplies by itself to give 3, about $1.732$. Notice $\tan 45^{\circ }=1$: at $45^{\circ }$ the height equals the distance — a fact used directly in Example 4 of the parent note.

9. Two ideas that make problems solvable

• Complementary angles (2.4.03-Complementary-angles): the two non-right angles of a right triangle add to $90^{\circ }$, because all three angles sum to $180^{\circ }$ and one is already $90^{\circ }$. This lets you swap between the two viewpoints of the same triangle.
• Two equations, two unknowns: when a problem gives two angles (like "walk 20 m closer, angle changes"), you write one $\tan$ equation per position and combine them to eliminate the unknown distance. This is the algebra behind Examples 3 and 4.

Where all of this gets used in the real world: 2.5.02-Surveying-and-navigation.

Prerequisite map

How to read this diagram: each box is one idea from this page. An arrow $X\to Y$ means "you need $X$ firmly in place before $Y$ makes sense". Follow the arrows top-to-bottom and you are walking the exact order of the sections above; every path eventually funnels into the bottom box — the actual topic you are training for.

Equipment checklist

Cover the right side and answer aloud; reveal to check.