Question

similar figures
in exercises 1–8, the figures are similar. find each missing measure.
1.
triangle abc with ab = 4m, ac = 12m; triangle def with de = 6m, df = xm
2.
triangle ghi with gh = 9in, hi = 10in; triangle jkl with jk = xin, kl = 30in
3.
pentagon abcde with ab = 9km, ae = 11km; pentagon wxyz u with uz = 44km, uw = xkm
4.
triangle tsr with ts = xft, tr = 8ft; triangle vuw with vw = 19ft, uw = 38ft
5.
rectangle pqro with po = 3in, or = 6in; rectangle tsuv with ts = 32in, tu = xin
6.
parallelogram mnop with mn = 10m, no = xm; parallelogram wxyz with wx = 24m, xy = 36m
7.
triangle jkl with lm = 7m, jm = 8m, jn = 36m; triangle jkn with kn = xm
8.
triangle with base 63yd, side xyd; smaller triangle with base 15yd, side 12yd

1. geometry triangle abc is similar to triangle def. what is the value of \\(\overline{bc}\\) if \\(\overline{ef}\\) is 36 feet, \\(\overline{ac}\\) is 7 feet, and \\(\overline{df}\\) is 28 feet?
2. geometry quadrilateral rstu is similar to quadrilateral lmno. what is the value of \\(\overline{lo}\\) if \\(\overline{ru}\\) is 6 inches, \\(\overline{lm}\\) is 45 inches, and \\(\overline{rs}\\) is 9 inches?
3. quilts a woman sews similar quilts for her daughter and her daughter’s doll. if the daughter’s quilt has a length of 2 yards and a width of 1 yard, and the doll’s quilt has a length of \\(\frac{1}{2}\\) yard, what is the width of the doll’s quilt?

Explanation:

Exercise 1

Step1: Set up similarity proportion

$\frac{4}{6} = \frac{12}{x}$

Step2: Cross-multiply to solve for $x$

$4x = 6 \times 12$
$4x = 72$
$x = \frac{72}{4} = 18$

Exercise 2

Step1: Set up similarity proportion

$\frac{9}{x} = \frac{10}{30}$

Step2: Simplify and solve for $x$

$\frac{9}{x} = \frac{1}{3}$
$x = 9 \times 3 = 27$

Exercise 3

Step1: Set up similarity proportion

$\frac{11}{44} = \frac{9}{x}$

Step2: Simplify and solve for $x$

$\frac{1}{4} = \frac{9}{x}$
$x = 9 \times 4 = 36$

Exercise 4

Step1: Set up similarity proportion

$\frac{x}{19} = \frac{8}{38}$

Step2: Simplify and solve for $x$

$\frac{x}{19} = \frac{4}{19}$
$x = 4$

Exercise 5

Step1: Set up similarity proportion

$\frac{3}{x} = \frac{6}{32}$

Step2: Simplify and solve for $x$

$\frac{3}{x} = \frac{3}{16}$
$x = 16$

Exercise 6

Step1: Set up similarity proportion

$\frac{10}{24} = \frac{x}{36}$

Step2: Cross-multiply to solve for $x$

$24x = 10 \times 36$
$24x = 360$
$x = \frac{360}{24} = 15$

Exercise 7

Step1: Set up similarity proportion

$\frac{8}{36} = \frac{7}{x}$

Step2: Simplify and solve for $x$

$\frac{2}{9} = \frac{7}{x}$
$2x = 63$
$x = \frac{63}{2} = 31.5$

Exercise 8

Step1: Set up similarity proportion

$\frac{12}{63} = \frac{15}{x}$

Step2: Cross-multiply to solve for $x$

$12x = 63 \times 15$
$12x = 945$
$x = \frac{945}{12} = 78.75$

Exercise 9

Step1: Set up similarity proportion

$\frac{BC}{EF} = \frac{AC}{DF}$
$\frac{BC}{36} = \frac{7}{28}$

Step2: Simplify and solve for $BC$

$\frac{BC}{36} = \frac{1}{4}$
$BC = \frac{36}{4} = 9$

Exercise 10

Step1: Set up similarity proportion

$\frac{RS}{LM} = \frac{RU}{LO}$
$\frac{9}{45} = \frac{6}{LO}$

Step2: Simplify and solve for $LO$

$\frac{1}{5} = \frac{6}{LO}$
$LO = 6 \times 5 = 30$

Exercise 11

Step1: Set up similarity proportion

$\frac{\text{Doll's length}}{\text{Daughter's length}} = \frac{\text{Doll's width}}{\text{Daughter's width}}$
$\frac{\frac{1}{2}}{2} = \frac{w}{1}$

Step2: Simplify and solve for $w$

$\frac{1}{4} = w$

Answer:

1. $18$ m
2. $27$ in
3. $36$ km
4. $4$ ft
5. $16$ in
6. $15$ m
7. $31.5$ m
8. $78.75$ yd
9. $9$ feet
10. $30$ inches
11. $\frac{1}{4}$ yard