Question

directions

1. all figures below are similar. find the value of each missing variable using any method. show all work.

in the diagram, defg ~ pqrs.
(diagrams of similar figures with labeled sides and angles, and variables x, y, h to find)
(another set of similar triangle diagrams with labeled sides and variable x to find)
(diagram of a person and a flagpole with labeled distances and height h to find)

Explanation:

Step1: Analyze Similar Polygons (DEFG ~ PQRS)

For similar polygons, corresponding sides are proportional. First, identify corresponding sides. In DEFG, side DE = 36, EF = 24, FG = 56, GD = 48. In PQRS, side PQ = x, QR = 12, RS = y, SP = 24. The right angles are at D and P, so GD (48) corresponds to SP (24), EF (24) corresponds to PQ? Wait, no, let's find the scale factor. GD = 48, SP = 24, so scale factor from DEFG to PQRS is \( \frac{24}{48} = \frac{1}{2} \).

Step2: Find x (PQ)

EF in DEFG is 24, which should correspond to PQ in PQRS? Wait, no, DE is 36, let's check another side. GD = 48, SP = 24 (scale factor 1/2). So DE (36) corresponds to QR (12)? Wait, no, maybe DE corresponds to PQ? Wait, no, let's list the sides:

DEFG: D(right angle), E, F, G. Sides: DE = 36, EF = 24, FG = 56, GD = 48.

PQRS: P(right angle), Q, R, S. Sides: PQ = x, QR = 12, RS = y, SP = 24.

So corresponding sides: GD (48) ↔ SP (24) (scale 1/2), DE (36) ↔ PQ (x)? Wait, no, GD is vertical, SP is vertical. So GD (48) and SP (24) are corresponding, so scale factor is 24/48 = 1/2. Then DE (36) corresponds to QR (12)? 36*(1/2)=18, no, QR is 12. Wait, maybe EF (24) corresponds to SP (24)? No, SP is 24, EF is 24. Wait, maybe I mixed up. Let's do it properly.

Since DEFG ~ PQRS, the order of the letters matters: D↔P, E↔Q, F↔R, G↔S. So DE corresponds to PQ, EF corresponds to QR, FG corresponds to RS, GD corresponds to SP.

So DE = 36, PQ = x; EF = 24, QR = 12; FG = 56, RS = y; GD = 48, SP = 24.

Check the ratio of EF to QR: 24/12 = 2. GD to SP: 48/24 = 2. So scale factor from DEFG to PQRS is 1/2? Wait, no, EF is 24, QR is 12, so 12/24 = 1/2. So scale factor is 1/2. Then DE (36) * 1/2 = 18? But QR is 12, no. Wait, maybe DE corresponds to QR? 36*(1/2)=18, no. Wait, EF is 24, QR is 12 (24*1/2=12), correct. GD is 48, SP is 24 (48*1/2=24), correct. So DE (36) corresponds to PQ (x)? Wait, no, DE is adjacent to D, PQ is adjacent to P. So DE (36) and PQ (x) should have ratio 1/2? No, EF (24) and QR (12) have ratio 1/2, GD (48) and SP (24) have ratio 1/2. So DE (36) and PQ (x) should have ratio 1/2? Wait, no, maybe DE corresponds to RS? No, let's do PQ. Wait, maybe I made a mistake. Let's take another pair: FG = 56, RS = y. Since scale factor is 1/2, y = 56*(1/2) = 28? Wait, no, SP is 24, GD is 48, so SP = GD*(1/2). So RS = FG*(1/2) = 56*(1/2) = 28. PQ = DE*(1/2) = 36*(1/2) = 18. Wait, but QR is 12, EF is 24, 24*(1/2)=12, correct. So x (PQ) = 18, y (RS) = 28, and angle x (at Q) is equal to angle E (122°)? Wait, no, angle at Q: since P is right angle, and DEFG has angle at E as 122°, so angle at Q (x°) should be equal to angle at E? Wait, no, D is right angle (90°), P is right angle (90°). Angle at E is 122°, so angle at Q (x°) should be equal to angle at E? Wait, no, in DEFG, angles: D=90°, E=122°, F=?, G=?. In PQRS, angles: P=90°, Q=x°, R=?, S=?. Since similar, corresponding angles are equal. So angle at E (122°) corresponds to angle at Q (x°), so x°=122°? Wait, no, maybe angle at F corresponds to angle at R, angle at G corresponds to angle at S. Wait, maybe I messed up the angle. Let's check the first triangle (top right):

Triangle ADE and ABC? Wait, the top right figure: triangle with A, D, B, E, C. AD=3, DB=9, so AB=12. AE=5, EC=x, so AC=5+x. DE=y, BC=16. Since triangles ADE and ABC are similar (because DE || BC, by basic proportionality theorem or AA similarity), so AD/AB = AE/AC = DE/BC.

AD=3, AB=3+9=12. AE=5, AC=5+x. DE=y, BC=16.

So AD/AB = 3/12 = 1/4. Then AE/AC = 5/(5+x) = 1/4 → 5*4 = 5+x → 20=5+x → x=15.

DE/BC = y/16 = 1/4 → y=4.

Now the bottom left triangle: Triangle ABC and DBC? Wait, AB=14+8=22? No, AB=14, BD=8, so AD=AB+BD=22? Wait, no, the triangle: A, B, D; A, C, E. BC is parallel to DE? So triangle ABC ~ triangle ADE. AB=14, AD=14+8=22? Wait, no, the sides: AB=14, BD=8, so AD=AB + BD=22? Wait, BC=10, DE=x. So AB/AD = BC/DE → 14/22 = 10/x → 14x=220 → x=220/14=110/7≈15.71? Wait, maybe AB=14, AD=14+8=22? No, maybe the other way: AB=14, BD=8, so AD=AB + BD=22? Wait, the triangle: A, B, D; A, C, E. BC is a midline? No, BC=10, DE=x. So AB/AD = BC/DE. AB=14, AD=14+8=22? Wait, no, maybe AB=14, BD=8, so AD=AB + BD=22? Wait, the problem says "all figures are similar". Let's take the flagpole problem: person height 6ft, shadow 10ft; flagpole shadow 10+12=22ft. So 6/10 = h/22 → h= (6*22)/10=13.2ft.

But let's focus on the first similar polygons (DEFG ~ PQRS):

Scale factor: GD=48, SP=24 → scale 24/48=1/2.

DE=36, so PQ (x)=36*(1/2)=18.

EF=24, QR=12 → 24*(1/2)=12, correct.

FG=56, so RS (y)=56*(1/2)=28.

Angle at Q: angle E is 122°, so angle Q (x°)=122° (corresponding angles in similar polygons are equal).

Top right triangle (ADE ~ ABC):

AD=3, AB=3+9=12.

AE=5, AC=5+x.

DE=y, BC=16.

Similarity ratio: AD/AB=3/12=1/4.

So AE/AC=5/(5+x)=1/4 → 20=5+x → x=15.

DE/BC=y/16=1/4 → y=4.

Bottom left triangle (ABC ~ ADE):

AB=14, AD=14+8=22? Wait, no, AB=14, BD=8, so AD=AB + BD=22? BC=10, DE=x.

Similarity ratio: AB/AD=14/22=7/11.

So BC/DE=10/x=7/11 → x=110/7≈15.71.

Flagpole problem:

Person: height 6ft, shadow 10ft.

Flagpole: shadow 10+12=22ft.

Let h be flagpole height.

6/10 = h/22 → h= (6*22)/10=13.2ft.

But the question is to find each missing variable. Let's start with the first similar polygons (DEFG ~ PQRS):

x (PQ) = 18, y (RS) = 28, angle x (at Q) = 122°? Wait, no, angle at E is 122°, so angle at Q (x°) is equal to angle at E, so x=122.

Top right triangle:

x=15, y=4.

Bottom left triangle:

x=110/7 (or 15.71).

Flagpole:

h=13.2ft.

But the problem says "each missing variable", so we need to solve each figure.

Let's take the first figure (DEFG ~ PQRS):

1. Find x (PQ):

Corresponding sides: DE (36) ↔ PQ (x)? Wait, no, earlier we saw scale factor 1/2 (SP=24, GD=48). So DE (36) * (1/2) = 18? Wait, EF (24) * (1/2)=12 (QR=12), correct. So DE (36) corresponds to PQ (x), so x=36*(1/2)=18.

2. Find y (RS):

FG (56) corresponds to RS (y), so y=56*(1/2)=28.

3. Find angle x (at Q):

Angle at E is 122°, so angle at Q (x°) is equal to angle at E, so x=122.

Top right triangle (ADE ~ ABC):

AD=3, AB=3+9=12.

AE=5, AC=5+x.

DE=y, BC=16.

By similarity, AD/AB = AE/AC = DE/BC.

AD/AB = 3/12 = 1/4.

So AE/AC = 5/(5+x) = 1/4 → 5*4 = 5 + x → 20 = 5 + x → x=15.

DE/BC = y/16 = 1/4 → y=4.

Bottom left triangle (ABC ~ ADE):

AB=14, BD=8, so AD=AB + BD=22.

BC=10, DE=x.

By similarity, AB/AD = BC/DE → 14/22 = 10/x → 14x = 220 → x=220/14=110/7≈15.71.

Flagpole problem:

Person's height (T)=6ft, shadow=10ft.

Flagpole's shadow=10+12=22ft.

Let h be flagpole height.

6/10 = h/22 → h= (6×22)/10=13.2ft.

Since the problem is to find each missing variable, we can solve each figure. Let's pick one, say the top right triangle:

Explanation (Top Right Triangle):

Step1: Identify Similar Triangles

Triangles \( ADE \) and \( ABC \) are similar (by AA similarity, as \( DE \parallel BC \), so \( \angle ADE = \angle ABC \) and \( \angle AED = \angle ACB \)).

Step2: Set Up Proportions

The ratio of corresponding sides: \( \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC} \).

- \( AD = 3 \), \( AB = AD + DB = 3 + 9 = 12 \).
- \( AE = 5 \), \( AC = AE + EC = 5 + x \).
- \( DE = y \), \( BC = 16 \).

Step3: Solve for \( x \) (EC)

Using \( \frac{AD}{AB} = \frac{AE}{AC} \):
\( \frac{3}{12} = \frac{5}{5 + x} \)
Simplify \( \frac{1}{4} = \frac{5}{5 + x} \)
Cross - multiply: \( 5 + x = 5 \times 4 = 20 \)
Subtract 5: \( x = 20 - 5 = 15 \).

Step4: Solve for \( y \) (DE)

Using \( \frac{AD}{AB} = \frac{DE}{BC} \):
\( \frac{3}{12} = \frac{y}{16} \)
Simplify \( \frac{1}{4} = \frac{y}{16} \)
Multiply both sides by 16: \( y = \frac{16}{4} = 4 \).

(Top Right Triangle):

Step1: Identify Similar Triangles

Triangles \( ADE \) and \( ABC \) are similar (by AA similarity, as \( DE \parallel BC \), so \( \angle ADE = \angle ABC \) and \( \angle AED = \angle ACB \)).

Step2: Set Up Proportions

The ratio of corresponding sides: \( \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC} \).

• \( AD = 3 \), \( AB = AD + DB = 3 + 9 = 12 \).
• \( AE = 5 \), \( AC = AE + EC = 5 + x \).
• \( DE = y \), \( BC = 16 \).

Step3: Solve for \( x \) (EC)

Using \( \frac{AD}{AB} = \frac{AE}{AC} \):
\( \frac{3}{12} = \frac{5}{5 + x} \)
Simplify \( \frac{1}{4} = \frac{5}{5 + x} \)
Cross - multiply: \( 5 + x = 5 \times 4 = 20 \)
Subtract 5: \( x = 20 - 5 = 15 \).

Step4: Solve for \( y \) (DE)

Using \( \frac{AD}{AB} = \frac{DE}{BC} \):
\( \frac{3}{12} = \frac{y}{16} \)
Simplify \( \frac{1}{4} = \frac{y}{16} \)
Multiply both sides by 16: \( y = \frac{16}{4} = 4 \).

Answer:

For the top - right triangle: \( x = 15 \), \( y = 4 \)

(For other figures, follow similar steps:

• DEFG ~ PQRS: \( x = 18 \), \( y = 28 \), angle \( x = 122^\circ \)
• Bottom - left triangle: \( x=\frac{110}{7}\approx15.71 \)
• Flagpole: \( h = 13.2 \) ft)