Proportionality>Generalizing Ratios of Corresponding Sides in Similar Shapes(TEKS.Math.8.3.A)
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Texas 8th Grade Math › Proportionality>Generalizing Ratios of Corresponding Sides in Similar Shapes(TEKS.Math.8.3.A)
Triangle ABC has vertices A(2,4), B(6,8), C(4,12). Triangle DEF has vertices D(1,2), E(3,4), F(2,6). What is the scale factor of the dilation from triangle ABC to triangle DEF?
2
$1/2$
$3/2$
3
Explanation
Similar figures have corresponding sides in proportion. Compare a corresponding coordinate pair: from A(2,4) to D(1,2), each coordinate is multiplied by $1/2$, so every side length is also multiplied by $1/2$. Checking another pair, B(6,8) to E(3,4) shows the same factor. The scale factor (new ÷ original) is $1/2$, and all corresponding ratios match.
Rectangle PQRS has dimensions 5 by 12 units. After dilation, rectangle P'Q'R'S' has dimensions 15 by 36 units. What is the scale factor of the dilation from rectangle PQRS to rectangle P'Q'R'S'?
$1/3$
$12/5$
$5/12$
3
Explanation
For similar figures, the scale factor equals new length ÷ original length for any corresponding sides. Using the widths: $15 \div 5 = 3$. Using the heights: $36 \div 12 = 3$. Since both corresponding ratios are equal, the scale factor is 3.
Triangle GHI has side lengths 6, 9, and 12 units. Triangle JKL is similar to triangle GHI and has corresponding side lengths 8, 12, and 16 units. What is the ratio of corresponding sides (triangle JKL to triangle GHI)?
$4/3$
$3/4$
$2/3$
$3/2$
Explanation
Similarity means all corresponding side ratios are equal. Compare one pair: $8 \div 6 = 4/3$. Check others: $12 \div 9 = 4/3$ and $16 \div 12 = 4/3$. Since each new ÷ original ratio matches, the common ratio of corresponding sides (and the scale factor from GHI to JKL) is $4/3$.
Parallelogram ABCD has side lengths AB = 10 units and BC = 6 units. A'B'C'D' is a dilation of ABCD with corresponding side lengths A'B' = 15 units and B'C' = 9 units. What is the scale factor of the dilation from parallelogram ABCD to parallelogram A'B'C'D'?
$3/2$
$2/3$
$5/3$
$3/5$
Explanation
Use new ÷ original for corresponding sides. For AB: $15 \div 10 = 3/2$. For BC: $9 \div 6 = 3/2$. The equal ratios show the figures are similar by dilation, and the scale factor is $3/2$.