SAT-74: Congruent Triangles and Area/Perimeter Scaling

SAT-74: Congruent Triangles and Area/Perimeter Scaling

Description: Congruent figures are identical copies (same shape and size), while similar figures (SAT-73) share shape but differ in size by a scale factor. The SAT's favorite twist: when you scale a figure, perimeter and area do not grow the same way.

Congruent vs similar

• Congruent: same shape and same size — one fits exactly on the other. Scale factor = 1.
• Similar: same shape, sizes related by a scale factor k.

(Oʻzbekcha: kongruent — shakl ham, oʻlcham ham bir xil; oʻxshash — shakl bir xil, oʻlcham k marta farq qiladi.)

The scaling rule (the key idea)

If lengths scale by a factor k, then perimeter scales by k and area scales by k².

Perimeter is a length, so it grows in step with k. Area is two-dimensional, so it grows by k squared. (For volume, which you'll meet later, the factor is k³.) (Oʻzbekcha: uzunlik k marta, perimetr k marta, yuza esa k² marta oshadi.)

Worked Example 1 — scale the perimeter

A triangle has perimeter 12. A similar triangle is 3 times as big (k = 3). What is its perimeter?

• Perimeter scales by k: 12 × 3 = 36.

Worked Example 2 — scale the area

The original triangle has area 8. With k = 3, what is the new area?

• Area scales by k² = 3² = 9: 8 × 9 = 72.

(Oʻzbekcha: yuza k² marta oshadi, ya'ni 9 barobar.)

Worked Example 3 — work backward from areas

Two similar squares have areas 16 and 144. What is the ratio of their side lengths?

• Area ratio = 144 ÷ 16 = 9. Since area ratio = k², k = √9 = 3.
• So the sides are in ratio 3 : 1 (the bigger square's sides are 3× as long).

Trap: students multiply area by k instead of k². Always square the scale factor for area, and take the square root of an area ratio to get the length ratio. (Oʻzbekcha: yuza uchun k ni kvadratga koʻtaring; yuza nisbatidan uzunlik nisbatini olish uchun ildiz oling.)

Extending the idea to volume

The same logic continues into three dimensions. If lengths scale by k, then surface area scales by k² and volume scales by k³. So doubling every edge of a box (k = 2) multiplies its volume by 8, not 2. A simple way to remember the whole family: the exponent matches the number of dimensions — length is 1-D (k¹), area is 2-D (k²), volume is 3-D (k³). (Oʻzbekcha: uzunlik k¹, yuza k², hajm k³ marta oshadi — koʻrsatkich oʻlchamlar soniga teng.)

Why congruence criteria matter

Two triangles are congruent when enough matching parts are equal; the standard shortcuts are SSS (three sides), SAS (two sides and the angle between), ASA (two angles and the side between), and AAS. Notice that knowing all three angles alone (AAA) is not enough — that only proves similarity, because the triangles could be different sizes. This is the cleanest way to see the line between congruent (same size) and similar (same shape). (Oʻzbekcha: SSS, SAS, ASA, AAS — kongruentlik belgilari; faqat burchaklar (AAA) esa oʻxshashlikni beradi.)

Practice 1

A rectangle's sides are doubled (k = 2). By what factor does its area increase?

Practice 2

Two similar triangles have areas 25 and 100. If the smaller has perimeter 15, what is the larger's perimeter?

Key words — Kalit soʻzlar

• Congruent — kongruent (teng)
• Similar — oʻxshash
• Scale factor — oʻlchov koeffitsiyenti
• Perimeter — perimetr
• Area — yuza
• Ratio — nisbat
• Square (k²) — kvadrat (k²)
• Square root — kvadrat ildiz
• Dimension — oʻlcham

Summary

• Congruent = same shape and size; similar = same shape, scaled by k.
• Length and perimeter scale by k; area scales by k².
• From an area ratio, take the square root to get the length/perimeter ratio.