Chapter 8.2 — Exercise 8.1 — Congruency and Similarity

Does Moving a Shape Break Its Similarity?

Two congruent figures are always similar too (their corresponding sides are in a 1:1 ratio, which is still a valid proportion). The more interesting question is what survives when a similar pair gets moved: take ∆ABC ~ ∆PQR, then rotate ∆PQR through some angle, or flip it into its mirror image. In both cases, the ratio of corresponding sides stays exactly the same as before — rotating or flipping a shape never changes its side lengths or angles, so similarity (and congruence, where it applies) survives every rigid movement without exception.

Reading Off Corresponding Parts

Given ∆ABC ≅ ∆NMO, the order of the letters tells you everything: A corresponds to N, B corresponds to M, C corresponds to O. That gives congruent sides AB≅NM, BC≅MO, AC≅NO, and congruent angles ∠A≅∠N, ∠B≅∠M, ∠C≅∠O. No measuring is needed to answer this kind of question — the correspondence is entirely determined by the order the vertices were written in.

True or False, and Why

• Two 3 cm squares, one rotated 45°, are congruent. True — rotating a shape never changes its side lengths, so both squares remain fully congruent to one another regardless of how either one happens to be oriented.
• Any two right triangles with a 5 cm hypotenuse are congruent. False — the hypotenuse alone doesn't fix the other two sides; many different right triangles share a 5 cm hypotenuse (3-4-5 is only one option among infinitely many).
• Any two circles of radius 4 cm are congruent. True — equal radii is both necessary and sufficient for two circles to be congruent.
• Two equilateral triangles of side 4 cm, labelled ∆ABC and ∆LHN, are not congruent. False — the labels attached to the vertices are irrelevant; equal side lengths alone make them fully congruent, regardless of whatever the individual vertices happen to be called or labelled.
• A polygon's mirror image is congruent to the original. True — a reflection preserves every side length and every angle, satisfying the definition of congruence exactly.

Comparing Areas and Perimeters of Similar Rectangles

Rectangle ABCD (3 units by 2 units) and a similar rectangle EFGH (9 units by 6 units) share a scale ratio of 1:3 on both length and breadth. Their perimeters — 10 units and 30 units — are also in the ratio 1:3, matching the side ratio exactly. But their areas — 6 sq. units and 54 sq. units — are in the ratio 1:9, the square of the side ratio, not the ratio itself.

This distinction — perimeter scales linearly, area scales quadratically — is easy to state and easy to forget under pressure, so it's worth checking against the concrete numbers above rather than trusting the rule from memory alone.

Measuring a Row of Pillars With One Known Height

Seven pillars stand 1 m apart, holding up a slanted girder. The tallest, 7 m from the start, measures 10.5 m. Every shorter pillar forms a triangle similar to the full triangle formed by the tallest one, sharing the same base angle — so height is directly proportional to distance from the start: height ÷ distance = 10.5 ÷ 7 = 1.5 for every single pillar.

• 1st pillar (1 m along): 1 × 1.5 = 1.5 m
• 2nd pillar (2 m along): 2 × 1.5 = 3 m
• 3rd pillar (3 m along): 3 × 1.5 = 4.5 m
• 4th pillar (4 m along): 4 × 1.5 = 6 m
• 5th pillar (5 m along): 5 × 1.5 = 7.5 m
• 6th pillar (6 m along): 6 × 1.5 = 9 m

Every height in this sequence is a whole-number multiple of the same constant ratio — a direct payoff of recognising similar triangles hiding inside what looks, at first glance, like a purely practical measurement problem.

Estimating a Building's Height Using a Pole

Standing 5 m from a 3 m pole, the tip of the pole lines up exactly with the top of a building 10 m further away. The line of sight forms two similar triangles: a small one (observer to pole) and a large one (observer to building), sharing the same angle at the observer's position. With the observer 5 m from the pole and 5+10 = 15 m from the building: BC/MN = AB/AM gives 3/MN = 5/15, so MN = 3 × 3 = 9 m. This "shadow method" — using a known short height and its line of sight to estimate an inaccessible tall one — is one of the oldest practical applications of similar triangles, long predating any formal definition of similarity, and it works equally well with a person's own shadow standing in for the pole.

Verifying a Dilation by Measuring

A rectangle with sides 3 and 2 units, dilated by a scale factor of 3, produces a rectangle with sides 9 and 6 units. Checking every corresponding ratio — 3/9, 2/6, 3/9, 2/6 — all four reduce to 1/3, confirming the two rectangles are genuinely similar, exactly as the dilation construction guarantees. This kind of direct check — measuring every side and computing every ratio, rather than trusting the scale factor alone — is what actually demonstrates similarity, instead of merely asserting it. The angles, too, would need checking in a general dilation, though for a rectangle the right angles are already guaranteed by definition, leaving only the side ratios genuinely in question.

Why the "Any Two Right Triangles" Statement Fails

It's worth lingering on why a shared hypotenuse alone can't guarantee congruence, since the reasoning generalises usefully. A right triangle with hypotenuse 5 cm could have legs 3 and 4 (the familiar 3-4-5 triangle), but it could equally have legs of roughly 2 and 4.58, or 1 and 4.9, or any other pair whose squares add to 25 — infinitely many valid combinations, each producing a differently shaped (non-congruent) triangle. A single measurement, even a significant one like the hypotenuse, is rarely enough to pin down an entire triangle on its own; that's exactly why congruence criteria in geometry always specify a combination of several measurements taken together, never a single one in isolation.

Scaling Ratios That Reappear in Mensuration

The area-versus-perimeter scaling distinction from this exercise reappears throughout mensuration in later chapters, wherever similar shapes of different sizes are compared. The rigid-transformation reasoning here — that rotating or flipping preserves both congruence and similarity — connects directly to Exercise 8.2, which studies what happens when a figure is compared to a transformed version of itself rather than to a separate figure. For the definitions this exercise assumes throughout, revisit the Introduction to Similar Figures before continuing further on into the next one.