Question

directions: the pairs of polygons below are similar. give the scale factor of figure a to figure b.
2.
figure of rectangle a (2 by 4) and rectangle b (8 by 16)
3.
figure of triangle a (sides 10, 15, 18) and triangle b (sides 4, 6, 7.2)

1. if the scale factor of figure a to figure b is 4:5, find the value of x.

figure of triangle a (side x) and triangle b (side 15)

1. if the scale factor of figure a to figure b is 7:2, find the perimeter of figure a.

figure of trapezoid a and trapezoid b (sides 12, 5, 9, 9)

1. if δabc ~ δdec, find the values of x and y.

figure of right triangle abc (sides 6, 10, x) and right triangle dec (sides 21, 28, y)

1. if δjkl ~ δnmp, find the value of x.

figure of triangle jkl (side 49, 9x + 1) and triangle nmp (side x + 5, 14)

1. if δdgh ~ δdef, find the value of x.

figure of triangle dgh (side 52, x + 3) and triangle def (side 91, 2x - 1)

Explanation:

Step1: (Problem 2) Match corresponding sides

Scale factor = $\frac{\text{Side of A}}{\text{Side of B}} = \frac{2}{16} = \frac{1}{8}$ or $\frac{4}{8} = \frac{1}{2}$
Correction: $\frac{2}{16} = \frac{1}{8}$ is wrong, correct is $\frac{2}{16} = \frac{1}{8}$ no, wait $\frac{4}{8} = \frac{1}{2}$, $\frac{2}{16} = \frac{1}{8}$ no, wait similar rectangles: A is 4x2, B is 8x16. So $\frac{4}{8} = \frac{1}{2}$, $\frac{2}{16} = \frac{1}{8}$ no, that can't be. Oh wait, no, I mixed up. Scale factor A to B is $\frac{\text{Side of B}}{\text{Side of A}}$? No, directions say scale factor of figure A to figure B, so $\frac{\text{Length in A}}{\text{Length in B}}$. Wait 4/8 = 1/2, 2/16 = 1/8, that's a problem. Oh no, I read the numbers wrong. Figure A is 4 (length) and 2 (width), Figure B is 8 (length) and 16 (width)? No, that can't be similar. Wait no, Figure B's length is 8, width is 16? No, similar rectangles must have proportional sides. So 4/8 = 2/4, but 16 is wrong. Oh wait, no, I misread: Figure B's width is 16, length is 8? Then 4/8 = 1/2, 2/16 = 1/8, that's not proportional. Wait no, maybe Figure A is 2 (length) and 4 (width)? No, the labels: A has sides 2 and 4, B has 8 and 16. Oh! 4/8 = 1/2, 2/4 = 1/2, no, B is 8 and 16, so 4/8 = 1/2, 2/8? No, no, similar polygons have corresponding sides proportional. So 4 corresponds to 8, 2 corresponds to 16? No, that's not proportional. Wait no, I flipped: scale factor A to B is $\frac{\text{Side of B}}{\text{Side of A}}$. So 8/4 = 2, 16/2 = 8, that's not right. Oh wait, no, the rectangle B has sides 8 and 16, so 8/16 = 1/2, rectangle A has 4/2 = 2, that's inverse. Oh! I see, I mixed up the corresponding sides. A is 2 (height) and 4 (base), B is 16 (height) and 8 (base)? No, that would be 2/16 = 1/8, 4/8 = 1/2, not proportional. Wait no, the problem says they are similar, so my reading is wrong. Oh! Figure B's height is 16, base is 8, so 8/16 = 1/2. Figure A's base is 4, height is 2, 4/2 = 2. Oh! Wait, scale factor A to B is $\frac{\text{Side of A}}{\text{Side of B}} = \frac{4}{8} = \frac{1}{2}$, and $\frac{2}{16} = \frac{1}{8}$? No, that can't be. Wait no, I think I read the numbers wrong. Figure A is 2 (base) and 4 (height), Figure B is 8 (base) and 16 (height). Then $\frac{2}{8} = \frac{1}{4}$, $\frac{4}{16} = \frac{1}{4}$. Yes! That's right, I mixed up the labels. So Step1 for Problem2: Corresponding sides: 2→8, 4→16. Scale factor = $\frac{2}{8} = \frac{1}{4}$.

Step2: (Problem3) Match corresponding sides

Scale factor = $\frac{\text{Side of A}}{\text{Side of B}} = \frac{10}{4} = \frac{5}{2}$, or $\frac{15}{7.2} = \frac{150}{72} = \frac{25}{12}$? No, wait 10 corresponds to 6, 15 corresponds to 7.2, 18 corresponds to 4? No, 10/6 = 5/3, 15/7.2 = 150/72 = 25/12, no. Wait 10 corresponds to 4, 15 corresponds to 7.2, 18 corresponds to 6: 10/4 = 5/2, 15/7.2 = 150/72 = 25/12, no. Wait 10/6 = 5/3, 18/4 = 9/2, no. Wait 10/4 = 2.5, 15/7.2 = 2.083, no. Wait 15/7.2 = 25/12, 10/4.8 = 25/12, but 4 is there. Oh! I see, 10 corresponds to 6, 18 corresponds to 4? No, 10/6 = 5/3, 18/4 = 9/2, no. Wait the angles: A has angles marked, B has corresponding angles. So side 10 in A corresponds to 4 in B? No, 15 in A corresponds to 7.2 in B, 18 in A corresponds to 6 in B. 18/6 = 3, 15/7.2 = 2.083, no. Wait 15/7.2 = 25/12, 10/4.8 = 25/12, but 4 is there. Oh! 10/6 = 5/3, 15/9 = 5/3, but 9 isn't there. Wait 7.2*5/3 = 12, no. Wait 6*3=18, 4*3=12, no. Wait 18/6=3, 15/7.2=2.083, no. Wait I think I got it backwards: scale factor A to B is $\frac{\text{Side of B}}{\text{Side of A}}$. 6/10 = 3/5, 7.2/15 = 12/25, no. 4/18 = 2/9, no. Wait 7.2/15 = 0.48, 6/10=0.6, 4/18≈0.222. No, that can't be. Wait no, 15/7.2 = 2.083, 10/4.8=2.083, 18/8.64=2.083. Oh! I misread B's sides: B has 6, 4, 7.2. So 10/6=5/3, 18/4=9/2, no. Wait 10/4=2.5, 15/7.2=2.083, no. Wait 18/7.2=2.5, 10/4=2.5, 15/6=2.5. Yes! That's it. Corresponding sides: 10↔4, 18↔7.2, 15↔6. So 10/4=2.5=5/2, 18/7.2=2.5=5/2, 15/6=2.5=5/2. So scale factor A to B is $\frac{5}{2}$.

Step3: (Problem4) Set up proportion

$\frac{4}{5} = \frac{x}{15}$
Solve for x: $x = 15 \times \frac{4}{5} = 12$

Step4: (Problem5) Find perimeter of B first

Perimeter of B = $12 + 9 + 9 + 5 = 35$
Scale factor A to B is 7:2, so $\frac{\text{Perimeter of A}}{\text{Perimeter of B}} = \frac{7}{2}$
Perimeter of A = $35 \times \frac{7}{2} = 122.5 = \frac{245}{2}$

Step5: (Problem6) Find scale factor

$\triangle ABC \sim \triangle DEC$, so scale factor = $\frac{AB}{DE} = \frac{6}{21} = \frac{2}{7}$
Solve for x: $\frac{10}{28} = \frac{x}{28}$? No, $\frac{AC}{DC} = \frac{x}{28} = \frac{2}{7}$, so $x = 28 \times \frac{2}{7} = 8$
Solve for y: $\frac{BC}{EC} = \frac{10}{y} = \frac{2}{7}$, so $y = 10 \times \frac{7}{2} = 35$

Step6: (Problem7) Set up proportion

$\triangle JKL \sim \triangle NMP$, so $\frac{JK}{NM} = \frac{JL}{NP}$
$\frac{49}{14} = \frac{9x+1}{x+5}$
$3.5 = \frac{9x+1}{x+5}$
$3.5(x+5) = 9x+1$
$3.5x + 17.5 = 9x + 1$
$17.5 - 1 = 9x - 3.5x$
$16.5 = 5.5x$
$x = 3$

Step7: (Problem8) Find scale factor

$\triangle DGH \sim \triangle DEF$, so scale factor = $\frac{DG}{DE} = \frac{52}{91} = \frac{4}{7}$
Set up proportion: $\frac{x+3}{2x-1} = \frac{4}{7}$
$7(x+3) = 4(2x-1)$
$7x + 21 = 8x - 4$
$21 + 4 = 8x - 7x$
$x = 25$

Answer:

1. $\frac{1}{4}$
2. $\frac{5}{2}$
3. $12$
4. $\frac{245}{2}$ or $122.5$
5. $x=8$, $y=35$
6. $3$
7. $25$