8.5 Scale Drawings, Scale Factor & Similar Figures

8.5 Scale Drawings, Scale Factor & Similar Figures

Quick Answer: A scale factor $k$ is the ratio of a length in the image to the corresponding length in the original. Applying it multiplies every length by $k$ and leaves every angle unchanged. The consequence students most often miss: lengths scale by $k$, areas by $k^2$, and volumes by $k^3$. Two figures are similar when corresponding angles are congruent and corresponding sides are proportional; they are congruent when $k = 1$. A scale on a map or blueprint is a conversion ratio - set the proportion up with matching units in matching positions and the arithmetic takes care of itself.

---

Where This Sits in the Blueprint

Scale sits at the junction of geometry and proportional reasoning, which is why it shows up in two different reporting categories on the grade 7 test.

---

Scale Factor: What Changes and What Does Not

$k = \frac{\text{length in the image}}{\text{corresponding length in the original}}$

Preserved by scaling: angle measures, parallelism, shape, ratios of lengths within the figure. Changed by scaling: every length, the perimeter, the area, the volume.

That first list is the reason similar figures "look the same." A triangle scaled by 3 has the same three angles it always had; only the ruler changed.

---

The $k$, $k^2$, $k^3$ Rule

This is the highest-value idea in the section and the one most reliably tested.

$\text{lengths} \times k \qquad \text{areas} \times k^2 \qquad \text{volumes} \times k^3$

Why. Area is a product of two lengths. Scale both by $k$ and the product scales by $k \times k = k^2$. Volume is a product of three lengths, so it scales by $k^3$.

Worked example. A rectangle measures 4 cm by 6 cm, with perimeter 20 cm and area 24 cm². Apply a scale factor of 3.

Check the area directly: $12 \times 18 = 216$, and $24 \times 9 = 216$. They agree.

[!IMPORTANT] Perimeter scales like a length, not like an area. Perimeter is a sum of lengths, so it scales by $k$. Only area scales by $k^2$. Students who apply $k^2$ to perimeter get a right idea in the wrong place.

Working backward from area. If a scaled figure has 25 times the area of the original, the scale factor is $\sqrt{25} = 5$, not 25. If a scaled solid has 64 times the volume, the scale factor is $\sqrt[3]{64} = 4$.

---

Similar Figures

Two figures are similar ($\sim$) when:

1. Corresponding angles are congruent, and
2. Corresponding sides are proportional.

Both conditions are required. A rectangle 2 by 3 and a rectangle 4 by 5 have four right angles each, so the angles match - but $\frac{4}{2} = 2$ while $\frac{5}{3} \approx 1.67$, so the sides are not proportional and the rectangles are not similar.

Triangles are the exception that makes life easy. If two angles of one triangle are congruent to two angles of another, the triangles are similar. The third angle follows automatically from the 180-degree sum, and proportional sides follow from that.

Naming Order Matters

Writing $\triangle ABC \sim \triangle DEF$ asserts specific correspondences: $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$. Therefore

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$

Setting up $\frac{AB}{EF}$ pairs sides that do not correspond, and the answer will be wrong even though the arithmetic is clean. Match the letters in the order they were given.

---

Reading and Using a Scale

A scale is written as a ratio: 1 in : 8 ft, 1 cm : 25 km, or 1 : 500. Treat it as a conversion ratio and build a proportion with matching units stacked.

Example. A blueprint uses 1 in : 8 ft. A hallway measures 3.25 in on the drawing. How long is the actual hallway?

$\frac{1 \text{ in}}{8 \text{ ft}} = \frac{3.25 \text{ in}}{x \text{ ft}} \quad \Rightarrow \quad x = 8 \times 3.25 = 26 \text{ ft}$

Example, reversed. The same blueprint must show a 60-foot wall. How long is the drawn segment?

$\frac{1 \text{ in}}{8 \text{ ft}} = \frac{x \text{ in}}{60 \text{ ft}} \quad \Rightarrow \quad x = \frac{60}{8} = 7.5 \text{ in}$

   drawing length     drawing length
   --------------  =  --------------      <- keep drawing over actual on BOTH sides
   actual length      actual length

   1 in                3.25 in
   -----      =        -------            <- units in matching positions
   8 ft                  x ft

Scale and area on a drawing. A patio drawn at 1 in : 4 ft measures 3 in by 5 in on paper, so its drawn area is 15 in². The actual patio is 12 ft by 20 ft, an area of 240 ft². Note that $240 \ne 15 \times 4$; it is $15 \times 4^2 = 240$. The $k^2$ rule governs here exactly as it does anywhere else.

---

Indirect Measurement with Similar Triangles

You cannot hold a tape measure to the top of a light pole, but a shadow gives you a similar triangle.

Setup. A 6-foot person casts a 4-foot shadow at the same moment a light pole casts a 22-foot shadow. Because the sun's rays strike both at the same angle and both objects stand vertically, the two triangles are similar.

$\frac{\text{person's height}}{\text{person's shadow}} = \frac{\text{pole's height}}{\text{pole's shadow}} \quad \Rightarrow \quad \frac{6}{4} = \frac{h}{22}$

$4h = 132 \quad \Rightarrow \quad h = 33 \text{ ft}$

Sanity check: the pole's shadow is 5.5 times the person's, so the pole should be 5.5 times the person's height, and $6 \times 5.5 = 33$. Consistent.

[!IMPORTANT] The measurements must be simultaneous. Shadow-based similarity depends on the sun being in the same position for both objects. A problem that gives one shadow at 9 a.m. and another at 3 p.m. is describing two different triangles, and the proportion does not hold.

---

Dilation on the Coordinate Plane

In grade 8, scaling becomes a transformation. A dilation centered at the origin with scale factor $k$ maps $(x, y) \rightarrow (kx, ky)$.

Dilation is the one transformation that does not preserve distance. Translations, reflections, and rotations are rigid motions producing congruent images; dilation produces a similar image. That distinction is the backbone of the grade 8 congruence-and-similarity benchmarks: two figures are congruent when a sequence of rigid motions maps one onto the other, and similar when a sequence of rigid motions and a dilation does.

---

Common Exam Traps & Misconceptions

[!WARNING]

Trap 1: Scaling area by $k$ instead of $k^2$

Doubling every side of a square does not double its area - it quadruples it. A 5 by 5 square has area 25; a 10 by 10 square has area 100.

[!WARNING]

Trap 2: Taking the scale factor straight from an area ratio

If a scaled figure has 9 times the area, the scale factor is 3, because $k^2 = 9$. Answer choices routinely offer 9 for exactly this reason.

[!WARNING]

Trap 3: Pairing non-corresponding sides

In $\triangle ABC \sim \triangle DEF$, side $AB$ corresponds to $DE$, not to $EF$. Rewrite the similarity statement above the proportion and match letter positions before writing any numbers.

[!WARNING]

Trap 4: Assuming equal angles alone make figures similar

Every rectangle has four right angles, yet rectangles are similar only when their side ratios match. For triangles, two congruent angles do suffice - but that shortcut belongs to triangles alone.

[!WARNING]

Trap 5: Inverting the scale ratio

With 1 in : 8 ft, a drawing measurement is multiplied by 8 to get the real length, and a real length is divided by 8 to get the drawing length. Before computing, ask whether the answer should be larger or smaller than the number you were given.