Reach the heights and distances you can't touch — with a shadow, a mirror, and a proportion.
Thales, the story goes, measured the Great Pyramid of Egypt by its shadow alone — never climbing a single step. You can do the same. The world is full of heights and widths you cannot climb or cross: a tree, a tower, a river. But each one quietly casts a similar triangle that you can measure — a shadow on the ground, a reflection in a mirror, a sight-line over a pole. Measure the small triangle you can reach, set up a proportion, and scale up to the answer. This last lesson of the stage is all method: turn a real scene into a figure, match the corresponding sides, and solve.
Everything here rests on one fact from 17.3: two triangles with the same angles are similar (AA), so their corresponding sides share one ratio. Nature keeps handing us pairs of equal angles for free — parallel sun rays, a mirror's equal bounce, a fixed line of sight. Spot the equal angles, name the similar triangles, write one proportion, and the unreachable length falls out.
Stand a person and a tall tree side by side in sunlight. At one instant the sun is so far away that its rays arrive parallel. So the ray grazing the top of the person's head and the ray grazing the top of the tree strike the ground at the same angle. Each object stands vertical, so each makes a right triangle with its own shadow — and the two right triangles share that equal sun-angle. Two equal angles is all we need: △ person ~ △ tree (AA).
Because the triangles are similar, corresponding sides are in proportion. Matching height to height and shadow to shadow,
person height : person-shadow = tree height : tree-shadow, that is personperson-shadow = treetree-shadow.
A person 1.5 m tall casts a 2 m shadow. At the very same moment a nearby tree casts an 8 m shadow. How tall is the tree?
Set up the proportion and solve: 1.52 = tree8 ⇒ tree = 1.5 × 8 ÷ 2 = 6 m. The person's height is ¾ of their shadow (1.5 = ¾ × 2), and the tree's height obeys the same ratio, so the tree is ¾ of 8 m = 6 m.
Both shadows must be measured at the same instant. The sun moves; an hour later the angle θ has changed and the person's shadow no longer matches the tree's. Measure together, or your two triangles aren't similar.
Lower the sun and every shadow grows; raise it and they shrink. The answer never changes — that is the whole point of using a proportion.
What if it's a cloudy day, or the tower casts no clean shadow? Lay a small mirror flat on the ground between you and the tower, and step back until you see the top of the tower reflected in it. Light obeys a simple law: it leaves a mirror at the same angle it arrives — the angle of incidence equals the angle of reflection. So the ray from the tower-top down to the mirror, and the ray from the mirror up to your eye, make equal angles with the ground.
Now look at the two right triangles standing on the ground: your eye, your feet, and the mirror form one; the tower-top, the tower-base, and the mirror form the other. Both have a right angle at the ground, and the equal bounce gives the equal angle at the mirror. Two pairs of equal angles ⇒ the triangles are similar (AA), so
eye height : your distance to M = tower height : M's distance to tower.
Your eye is 1.6 m above the ground. You stand 1.2 m from the mirror, and the mirror is 9 m from the foot of the tower. How tall is the tower?
1.61.2 = tower9 ⇒ tower = 1.6 × 9 ÷ 1.2 = 12 m.
Hold a pole (or your thumb at arm's length) so its tip just lines up with the tower-top along your line of sight. The pole and the tower hang from the same sight-line through your eye — that shared line of sight gives the equal angle, so the small pole-triangle and the big tower-triangle are similar. A known little triangle in your line of sight is always enough.
Move the mirror toward or away from the tower. You shuffle your feet so you still catch the top in it — both distances change together, and the proportion still gives 12 m.
Heights aren't the only thing out of reach. How wide is a river you cannot cross? You can't stretch a tape across the water — but you can build, entirely on your own bank, a small triangle similar to the big one that spans the river. Measure the small triangle's sides, find the scale factor, and the width follows.
Here is a clean staked-out layout. Stand at A on the near bank, directly across the water from a fixed landmark T — a tree. Turn 90° and walk along the bank, planting a stake at B and continuing to C, so that AB and BC are both measured. Now from C turn 90° and walk inland, away from the river, until your eye, the stake B, and the tree T all fall on one straight sight-line — stop at D. Two right triangles appear: the big △ ABT spans the river, the small △ CBD sits on your bank. The right angles at A and C are equal, and the angles at B are vertical angles (the sight-line crosses the bank-line), so they are equal too. Two equal angles ⇒ △ ABT ~ △ CBD (AA).
Matching the two triangles vertex for vertex (A↔C, B↔B, T↔D), corresponding sides give one proportion:
river width ATinland walk CD = big base ABsmall base CB.
You measure AB = 18 m and CB = 6 m (so the big base is 3× the small base), and the inland walk gives CD = 10 m. The scale factor is ABCB = 186 = 3, so the river width is
AT = CD × 3 = 10 × 3 = 30 m.
Never a foot in the water.
You can't reach the far length, but you can reach a triangle similar to the one that holds it. Measure the reachable triangle, read off the scale factor k, and multiply.
Shadows, mirrors, rivers — every one of these is the same four moves. Learn the recipe once and you can attack any "measure the unmeasurable" problem.
1. Draw the scene as a figure. Sketch the objects, the ground, the rays or sight-lines. Mark every length you know.
2. Find the two similar triangles. Name why they're similar — equal sun angle, the mirror's equal bounce, a shared line of sight — almost always AA.
3. Write the proportion. Match corresponding sides in the same order: height-to-height, base-to-base.
4. Solve and state the units. Cross-multiply, isolate the unknown, and write "m" (or "km") at the end.
Press Next to reveal one step at a time: draw it, spot the similar triangles, write the proportion, solve.
Match corresponding sides in the same order. Writing person : tree-shadow = tree : person-shadow flips the proportion and gives a wrong answer. Height goes over base, on both sides — height-to-height, base-to-base.
The whole stage comes home to one move: when you can't reach a length, find a similar triangle that you can measure, then scale.
A 1.6 m person casts a 2 m shadow. At the same moment a flagpole casts a 10 m shadow. How tall is the flagpole?
Similar right triangles (parallel sun rays ⇒ AA): 1.62 = h10, so h = 1.6 × 10 ÷ 2 = 8 m.
A mirror lies 1.5 m from you (eye height 1.5 m) and 12 m from the base of a wall. You see the top of the wall in it. How tall is the wall?
Equal-angle bounce ⇒ similar triangles: 1.51.5 = wall12. Since eye height equals your distance to the mirror, the ratio is 1, so wall = 12 m.
Two similar triangles model a river crossing: the small base is 4 m and the large base is 20 m; the small triangle's height is 3 m. The river width is the large height. Find it.
Scale factor k = 20 ÷ 4 = 5. River width = small height × k = 3 × 5 = 15 m.
At noon a 2 m pole casts a 0.8 m shadow. A tree's shadow is 6 m long. How tall is the tree?
20.8 = tree6, so tree = 2 × 6 ÷ 0.8 = 12 ÷ 0.8 = 15 m. (Each object's shadow is 0.4 of its height; 6 ÷ 0.4 = 15.)
State the four steps of the similarity recipe in order.
(1) Draw the scene as a figure; (2) find the two similar triangles and name why (AA); (3) write the proportion matching corresponding sides in the same order; (4) solve and state the units.
Explain why the two shadow-triangles in the sun are similar.
The sun is so far away that its rays are parallel, so they strike every shadow at the same angle. Both objects stand vertical, giving each a right angle. Two equal angles ⇒ similar by AA.
Six questions to lock it in. Tap the answer you think is right.
This closing lesson is pure application: it asks nothing new about similarity, only that students use the AA criterion (17.3) and the proportional-sides property (17.4) on real scenes. The big idea worth saying out loud is that nature keeps producing pairs of equal angles for free — parallel sun rays, a mirror's equal bounce of light, a fixed line of sight — and each pair of equal angles plus a shared right angle is enough for AA. Once two triangles are similar, a single proportion converts a small, reachable measurement into the unreachable answer.
The misconception to watch is flipping the proportion: a student who writes person : tree-shadow = tree : person-shadow has crossed the correspondence and will get a wrong number. Drill the habit of matching sides in the same order — height over base on both sides of the equation — and have them sanity-check the answer (a tree should come out taller than its person, not shorter). Two further slips: forgetting that the shadow method needs both objects measured at the same instant (the sun moves), and dropping units in the final line.
Common Core: G-SRT.B.5 (use congruence and similarity criteria to solve problems and prove relationships), G-MG.A.1 (use geometric shapes and measures to model real-world objects), and 7.G.A.1 / 7.RP (scale, ratio, and proportional reasoning).