Heights and Distances

Split at the eye-level horizontal

Compound figures stack two objects — statue on a pedestal, tower on a building, flagstaff on a tower. One horizontal line through the change point (where the pedestal meets the statue, where the building ends) splits the figure into two right triangles that share a base distance $d$.

The workhorse: difference of two tangents

From a ground point, the elevation of the pedestal top is $\alpha$ and of the statue top is $\beta$:

• Pedestal: $p = d\tan\alpha$.
• Whole: $p + s = d\tan\beta$.
• Statue: $s = d(\tan\beta - \tan\alpha)$.

The $\alpha = 45^\circ$ special case is everywhere: it fixes $d = p$ (the pedestal height IS the distance), so the statue is $p(\tan\beta - 1)$.

Tower on a building from a distance

Building $b$, tower $t$, elevation of building top $\alpha$, of tower top $\beta$ from distance $d$: $b = d\tan\alpha$, $b + t = d\tan\beta$, so $t = d(\tan\beta - \tan\alpha)$. Same equation, different furniture. When the observation point is instead on the roof (elevation of the tower top $\beta$, depression of the base $\alpha$), the depression fixes the distance: $d = \frac{b}{\tan\alpha}$, and the tower is $d\tan\beta$ — don't forget to ADD the building back.

The broken tree (and its cousins)

A tree breaks; the standing part $a$ and the broken part $L$ form a right triangle with the ground: $L$ is the hypotenuse, the distance $b$ where the top touches is one leg, $a$ the other. Given the touch distance $b$ and the ground angle $\theta$:

• standing part $a = b\tan\theta$
• broken part $L = \frac{b}{\cos\theta}$
• original height $= a + L = b\tan\theta + \frac{b}{\cos\theta}$ The original height reappears in "the tree is $H$ m tall — where does it touch?" questions: solve $\frac{b}{\cos\theta} = H - b\tan\theta$ or use the identity $L^2 = a^2 + b^2$.

Windows: two elevations, one structure

Two windows in a building are observed from a point: elevations $\alpha$ and $\beta$ ($\beta > \alpha$), window heights differ by $k$ m. Then $k = d(\tan\beta - \tan\alpha)$ — the same difference-of-tangents machinery, and the reverse direction (find $d$) needs only a division.

Quick revision

• Shared base $d$: upper piece $= d(\tan\beta - \tan\alpha)$.
• $45^\circ$ below ⇒ $d$ = lower piece's height.
• Broken tree: standing $b\tan\theta$, broken $\frac{b}{\cos\theta}$; add for the original.
• Roof observation: depression gives $d$; add the building height back.
• Options in factored surds $((\sqrt3-1)$, $(\sqrt3+1))$ — keep them factored.