Geometry · Scale & Similarity
Dilation & Scale Factor
Self-Study Worksheet — 20 Key Problems
Name: ___________________________
Date: _______________
Score: _____ / 20
Scale Factor (k)
→ \( k = \dfrac{\text{Image length}}{\text{Original length}} \)
→ \( k = \dfrac{\text{Image length}}{\text{Original length}} \)
k > 1 → ENLARGEMENT (bigger)
0 < k < 1 → REDUCTION (smaller)
0 < k < 1 → REDUCTION (smaller)
Center of Dilation
→ Fixed point. All lines pass through it.
→ Fixed point. All lines pass through it.
Area Ratio
→ \( k^2 \) | Volume → \( k^3 \)
→ \( k^2 \) | Volume → \( k^3 \)
Angle → NEVER changes after dilation
Negative k → Dilation + 180° rotation
CORE FORMULA
\[
k = \frac{A'B'}{AB} \qquad
A'(x,y) = \bigl(k\cdot x,\; k\cdot y\bigr) \text{ (center at origin)}
\]
§1 · Finding the Scale Factor ★☆☆
Triangle ABC has side \(AB = 4\) cm. After dilation, \(A'B' = 10\) cm. Find the scale factor.
\( k = \dfrac{A'B'}{AB} = \dfrac{10}{4} = 2.5 \) → Enlargement since \(k > 1\).
1.
Rectangle ABCD has length \(AB = 6\) cm. Its image \(A'B'C'D'\) has \(A'B' = 3\) cm. What is the scale factor? Is this an enlargement or reduction?
2.
\(\triangle PQR \sim \triangle P'Q'R'\). If \(PQ = 5\), \(QR = 7\), \(PR = 9\) and \(P'Q' = 15\), find \(Q'R'\) and \(P'R'\).
TRAP
Hint: All sides share the same scale factor!
3.
A photo is 4 cm × 6 cm. It is enlarged so the longer side becomes 18 cm. What is the scale factor? What is the new shorter side?
§2 · Dilation on the Coordinate Plane ★★☆
Point \(A(3, -2)\) is dilated with center \(O(0,0)\) and scale factor \(k = 3\). Find \(A'\).
\( A' = (3 \times 3,\; 3 \times {-2}) = (9,\; -6) \)
4.
Point \(B(-4, 6)\) is dilated with center at the origin and \(k = \dfrac{1}{2}\). Find \(B'\).
5.
\(\triangle DEF\) has vertices \(D(2,4)\), \(E(6,0)\), \(F(0,0)\). After dilation centered at the origin with \(k = 1.5\), write all three image vertices.
6.
Point \(C(5, 10)\) is dilated to \(C'(2, 4)\). The center of dilation is the origin. Find \(k\).
THINK
7.
Center of dilation is \((2, 3)\), \(k = 2\). Point \(P(5, 7)\) is dilated. Find \(P'\).
TRAP
Formula: \(P' = \bigl(c_x + k(x - c_x),\; c_y + k(y - c_y)\bigr)\)
§3 · Area & Perimeter Ratios ★★☆
A square has area \(16 \text{ cm}^2\). After dilation with \(k = 3\), what is the new area?
New area \(= k^2 \times 16 = 9 \times 16 = 144 \text{ cm}^2\)
Perimeter scales by \(k\). Area scales by \(k^2\). Volume scales by \(k^3\). Angles stay the SAME!
8.
A triangle has perimeter \(30\) cm and area \(36 \text{ cm}^2\). After dilation with \(k = 2\), find the new perimeter and new area.
9.
Two similar rectangles have areas \(25 \text{ cm}^2\) and \(100 \text{ cm}^2\). What is the scale factor from the smaller to the larger?
TRAP
Area ratio = \(k^2\), so first find \(k^2\), then take the square root!
10.
A circle has radius \(r = 5\) cm. After dilation with \(k = \dfrac{3}{5}\), find the new circumference. Leave in terms of \(\pi\).
§4 · Finding Missing Lengths in Similar Figures ★★☆
In \(\triangle ABC \sim \triangle DEF\): \(AB = 8, BC = 12, DE = 6\). Find \(EF\).
\( k = \dfrac{DE}{AB} = \dfrac{6}{8} = \dfrac{3}{4} \) →
\( EF = \dfrac{3}{4} \times 12 = 9 \)
11.
\(\triangle ABC \sim \triangle XYZ\). \(AB = 10,\; BC = 15,\; XY = 4\). Find \(YZ\).
12.
Two similar pentagons have perimeters \(20\) and \(35\). If one side of the smaller is \(4\), find the corresponding side of the larger.
13.
A 6-foot-tall person casts a shadow of 4 feet. At the same time, a tree casts a shadow of 22 feet. How tall is the tree?
REAL LIFE
§5 · Scale Factor from Coordinates ★★★
\(A(1,2) \to A'(3,6)\). Center of dilation is \(O(0,0)\). Find \(k\).
\( k = \dfrac{3}{1} = 3 \) (check: \(\dfrac{6}{2} = 3\) ✓)
14.
\(\triangle ABC\) has \(A(0,0), B(4,0), C(0,3)\). Its image has \(A'(0,0), B'(6,0)\). Find \(k\) and the coordinates of \(C'\).
15.
After a dilation centered at the origin, \(P(x, y) \to P'(5x, 5y)\). A point \(Q'\) in the image is at \((20, -15)\). What were the original coordinates of \(Q\)?
TRAP
Working backwards: divide by \(k\)!
§6 · Tricky & Mixed Concepts ★★★
16.
True or False? After dilation: (a) Angles change (b) Shape changes (c) Parallel lines stay parallel (d) Side lengths stay the same.
CONCEPT
17.
A square with side \(s = 4\) cm is dilated with \(k = -2\) centered at the origin. Describe the resulting image: Where is it? What are its dimensions?
NEGATIVE k
Negative scale factor = dilation + 180° rotation about the center.
18.
Two similar cylinders have volumes \(V_1 = 8 \text{ cm}^3\) and \(V_2 = 64 \text{ cm}^3\). Find the scale factor \(k\) and the ratio of their surface areas.
3D
Volume ratio = \(k^3\) → find \(k\) → Surface area ratio = \(k^2\)
19.
A map has a scale of \(1:50{,}000\). Two cities are \(7\) cm apart on the map. What is the real distance in kilometers?
REAL LIFE
\(1 \text{ km} = 100{,}000 \text{ cm}\)
20.
\(\triangle ABC\) is dilated to form \(\triangle A'B'C'\). \(A'B' = 3k + 1\) and \(AB = k + 5\). The scale factor is \(2\). Find the value of \(k\).
ALGEBRA
★ BOSS ★
Set up: \(\dfrac{A'B'}{AB} = 2\), then solve for \(k\).
🟢 FINAL CHEAT SHEET — EXPLAIN IN YOUR OWN WORDS
SCALE FACTOR
"How many times bigger/smaller is the image?"
"How many times bigger/smaller is the image?"
CENTER
"The fixed anchor point — never moves."
"The fixed anchor point — never moves."
k > 1 = stretch out | k < 1 = shrink in
SAME: angles, shape
DIFFERENT: lengths, area, volume
DIFFERENT: lengths, area, volume
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