Dilations & Scale Factors
Define "scale factor." If a figure is dilated with scale factor 2, how does a side length change?
Scale factor multiplies linear measures. A side length doubles (×2).
Write the coordinate rule for a dilation with center at the origin and scale factor 4.
Rule: (x, y) → (4x, 4y).
If a figure's side lengths are multiplied by 3 under dilation, by what factor does the perimeter change?
Perimeter multiplies by 3.
If ∆FUN is similar to ∆TIP, name one pair of corresponding sides.
FU ↔ TI, UN ↔ IP, or FN ↔ TP (depending on stated correspondence).
Who is the coolest coach?
Coach Padron
A door mat is similar to a rug. Each linear dimension of the mat is 1/3 the size of the rug. What is the ratio of the areas (mat : rug)?
Areas scale by the square of the linear scale: (1/3)^2 = 1/9. Ratio = 1:9 (mat : rug).
What transformation does the rule (x, y) → (−x, −y) represent? Does it preserve congruence?
This is a 180° rotation about the origin (equivalently a point reflection through the origin). It preserves congruence.
If a figure is reduced by scale factor 1/3, what happens to its perimeter and its area? Give the scale factors for both.
Perimeter scale factor = 1/3. Area scale factor = (1/3)^2 = 1/9.
Given similar triangles, explain how corresponding angles compare.
Corresponding angles are congruent (equal in measure).
Who do the Tigers Play today?
Coolidge
Triangle ABC is dilated from center P with scale factor 3 to form A'B'C'. If AB = 5 cm, find A'B'. If the perimeter of ABC is 24 cm, what is the perimeter of A'B'C'?
A'B' = 15 cm. Perimeter = 24 × 3 = 72 cm.
The rule (x, y) → (2x, 2y) is applied to a quadrilateral. If one vertex is (−5, 3), where will its image be?
Image at (−10, 6).
A triangle has side lengths 6, 8, 10 and area 24. After a dilation with scale factor 1/2, what are the new side lengths and new area?
New sides = 3, 4, 5. New area = 24 × (1/2)^2 = 24 × 1/4 = 6.
Triangle CAT is dilated about the origin to form ∆DOG. If C maps to D, A maps to O, and T maps to G, what is the scale factor if C(2,3) maps to D(6,9)?
Scale factor = 3 (since 2 → 6 and 3 → 9).
Who did the JV Tigers play yesterday?
Jonesboro
A trapezoid is reduced by a scale factor of 1/2. By what factor does its area change? Show work.
Area scale factor = (1/2)^2 = 1/4.
A polygon uses rule (x, y) → (ax, ay). If a = 0.25, describe whether this is an enlargement or reduction and explain how you know.
a = 0.25 is a reduction because 0 < a < 1; coordinates and linear measures shrink by factor 0.25.
A trapezoid's linear dimensions are multiplied by 5. If its original area was 12 cm^2, what is the new area? Explain reasoning.
Area scales by 5^2 = 25. New area = 12 × 25 = 300 cm^2.
A trapezoid has vertices F(−9, 6), G(−6, 9), H(−3, 9), I(0, 6). Dilate by 1/3 about origin. Give coordinates of F'G'H'I' and state whether this is an enlargement or reduction.
(x, y) → (1/3 x, 1/3 y). F'(−3, 2), G'(−2, 3), H'(−1, 3), I'(0, 2). This is a reduction.
Who do the Tigers play next week for homecoming?
Oakwood
A rectangle has area 48 cm^2. It is dilated by scale factor 3. What is the new area? Explain how you used the scale factor to find area.
New area = 48 × 3^2 = 48 × 9 = 432 cm^2.
Rectangle QRST has Q(−8, 5), R(−8, 7), S(−5, 7), T(−5, 5). Dilate by scale factor 4 about the origin. Write the rule, list coordinates of Q'R'S'T', and compute the new perimeter.
Rule: (x, y) → (4x, 4y). Q'(−32, 20), R'(−32, 28), S'(−20, 28), T'(−20, 20). Original side lengths: QR = 2, RS = 3, ST = 2, TQ = 3; original perimeter = 10. New perimeter = 4 × 10 = 40.
∆FED has coordinates F(1, 8), E(6, 11), D(8, 6). Dilate ∆FED by scale factor 1/2 about the origin to form ∆KIT. Give KIT coordinates, state whether this is an enlargement or reduction, and compare perimeters and areas (use numeric answers).
Rule: (x, y) → (1/2 x, 1/2 y). K = F' = (0.5, 4), I = E' = (3, 5.5), T = D' = (4, 3). This is a reduction (scale factor 1/2). Perimeter scales by 1/2; area scales by (1/2)^2 = 1/4. (Compute original perimeter and area from coordinates, then multiply by 1/2 and 1/4 respectively; verify with coordinate distance and area formulas.)
Given triangles with corresponding side ratios 2:5 (small : large), determine the perimeter and area scale factors. If the smaller triangle's perimeter is 18 and area is 24, find the larger triangle's perimeter and area.
Perimeter scale factor = 5/2. Area scale factor = (5/2)^2 = 25/4. Larger perimeter = 18 × 5/2 = 45. Larger area = 24 × 25/4 = 150.
When is Baby Bostyn due?
October 24th