similar figures—word problems
1. p and q are points on ab and bc respectively of δ abc. pq ∥ ac. ac = 18 cm, ap = 4 cm and pb = 12 cm. find pq.
2. the sides of a quadrilateral are 12′, 18′, 20′, and 16′. the longest side of a similar quadrilateral is 6′. find the remaining sides.
3. the corresponding sides of two similar figures are 12″ and 21″. find the ratio of their perimeters.
4. a triangle has sides of 6″, 4″ and 7″. the shortest side of a similar triangle is 12″. find the perimeter.
5. in δ abc, p is on ab and q is on ac. ∠apq = ∠c. if ab = 40′, ac = 32′, and cq = 6′, find pb.
6. in a 30° - 60° - 90° triangle, the shortest side is 15″. a similar triangle is four times as large. find the longest side of the similar triangle.
7. a field has the dimensions 375 yd, 425 yd, 275 yd and 300 yd. a plot plan is drawn to scale where 1″ represents 25 yd. find the dimensions of the plot plan.
8. in δ abc, points p and q are on ab and ac respectively. ∠aqp = ∠b. if ab = 24″, aq = 8″, cq = 22″, and pq = 12″, find ap, pb, and bc.
9. lines ab and cd intersect at e. ac is drawn parallel to bd. if ae = 9′, ec = 15′, and be = 12′, find de.
10. in δ abc, d is on ab and e is on ac. de ∥ bc. ac = 20 cm, ae = 6 cm, bc = 24 cm, and ad = 8 cm. find de and ab.

Question

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Identify similar triangles
Since $ PQ \parallel AC $, $ \triangle BPQ \sim \triangle BAC $ (by AA similarity criterion).
## Step2: Find the ratio of corresponding sides
$ AB = AP + PB = 4 + 12 = 16 \, \text{cm} $. The ratio of similarity $ k=\frac{PB}{AB}=\frac{12}{16}=\frac{3}{4} $.
## Step3: Find $ PQ $
Since the triangles are similar, $ \frac{PQ}{AC}=k $. So $ PQ = AC\times k = 18\times\frac{3}{4}=\frac{54}{4} = 13.5 \, \text{cm} $.
# Answer: $ 13.5 \, \text{cm} $

### Problem 2:
# Explanation:
## Step1: Find the scale factor
The longest side of the first quadrilateral is $ 20' $, and the longest side of the similar quadrilateral is $ 6' $. The scale factor $ k=\frac{6}{20}=\frac{3}{10} $.
## Step2: Find the remaining sides
- For the side $ 12' $: $ 12\times\frac{3}{10}=3.6' $
- For the side $ 18' $: $ 18\times\frac{3}{10}=5.4' $
- For the side $ 16' $: $ 16\times\frac{3}{10}=4.8' $
# Answer: $ 3.6' $, $ 5.4' $, $ 4.8' $

### Problem 3:
# Explanation:
## Step1: Recall the property of similar figures
The ratio of the perimeters of two similar figures is equal to the ratio of their corresponding sides.
## Step2: Simplify the ratio
The ratio of corresponding sides is $ \frac{12}{21}=\frac{4}{7} $. So the ratio of their perimeters is also $ \frac{4}{7} $.
# Answer: $ \frac{4}{7} $

### Problem 4:
# Explanation:
## Step1: Find the scale factor
The shortest side of the first triangle is $ 4'' $, and the shortest side of the similar triangle is $ 12'' $. The scale factor $ k = \frac{12}{4}=3 $.
## Step2: Find the other sides of the similar triangle
- Second side: $ 6\times3 = 18'' $
- Third side: $ 7\times3 = 21'' $
## Step3: Find the perimeter
Perimeter $ = 12 + 18 + 21 = 51'' $.
# Answer: $ 51'' $

### Problem 5:
# Explanation:
## Step1: Identify similar triangles
Since $ \angle APQ=\angle C $ and $ \angle A $ is common, $ \triangle APQ \sim \triangle ACB $ (by AA similarity).
## Step2: Set up the proportion
Let $ AP = x $, then $ PB = 40 - x $, and $ AQ = 32 - 6 = 26' $. So $ \frac{AP}{AC}=\frac{AQ}{AB} $, i.e., $ \frac{x}{32}=\frac{26}{40} $.
## Step3: Solve for $ x $
$ x=\frac{26\times32}{40}=\frac{832}{40}=20.8' $. Then $ PB = 40 - 20.8 = 19.2' $.
# Answer: $ 19.2' $

### Problem 6:
# Explanation:
## Step1: Recall the sides of a $ 30^\circ - 60^\circ - 90^\circ $ triangle
In a $ 30^\circ - 60^\circ - 90^\circ $ triangle, the sides are in the ratio $ 1:\sqrt{3}:2 $, where the shortest side (opposite $ 30^\circ $) is $ 1 $ part, and the longest side (hypotenuse) is $ 2 $ parts.
## Step2: Find the scale factor
The similar triangle is four times as large, so scale factor $ k = 4 $.
## Step3: Find the longest side
The shortest side of the original triangle is $ 15'' $, so the longest side of the original triangle is $ 30'' $. The longest side of the similar triangle is $ 30\times4 = 120'' $.
# Answer: $ 120'' $

### Problem 7:
# Explanation:
## Step1: Find the scale factor
The scale is $ 1'' $ represents $ 25 \, \text{yd} $, so scale factor $ k=\frac{1}{25} $.
## Step2: Find the dimensions of the plot plan
- For $ 375 \, \text{yd} $: $ \frac{375}{25}=15'' $
- For $ 425 \, \text{yd} $: $ \frac{425}{25}=17'' $
- For $ 275 \, \text{yd} $: $ \frac{275}{25}=11'' $
- For $ 300 \, \text{yd} $: $ \frac{300}{25}=12'' $
# Answer: $ 15'' $, $ 17'' $, $ 11'' $, $ 12'' $

### Problem 8:
# Explanation:
## Step1: Identify similar triangles
Since $ \angle AQP=\angle B $ and $ \angle A $ is common, $ \triangle AQP \sim \triangle ABC $ (by AA similarity).
## Step2: Find $ AC $
$ AC = AQ + CQ = 8 + 22 = 30'' $.
## Step3: Set up proportions
- For $ AP $ and $ AB $: $ \frac{AP}{AC}=\frac{AQ}{AB} $, let $ AP = x $, then $ \frac{x}{30}=\frac{8}{24} $, $ x = 10'' $. So $ PB = AB - AP = 24 - 10 = 14'' $.
- For $ BC $: $ \frac{PQ}{BC}=\frac{AQ}{AB} $, $ \frac{12}{BC}=\frac{8}{24} $, $ BC = 36'' $.
# Answer: $ AP = 10'' $, $ PB = 14'' $, $ BC = 36'' $

### Problem 9:
# Explanation:
## Step1: Identify similar triangles
Since $ AC \parallel BD $, $ \triangle AEC \sim \triangle BED $ (by AA similarity).
## Step2: Set up the proportion
$ \frac{AE}{BE}=\frac{EC}{DE} $, so $ \frac{9}{12}=\frac{15}{DE} $.
## Step3: Solve for $ DE $
$ 9\times DE = 12\times15 $, $ DE=\frac{180}{9}=20' $.
# Answer: $ 20' $

### Problem 10:
# Explanation:
## Step1: Identify similar triangles
Since $ DE \parallel BC $, $ \triangle ADE \sim \triangle ABC $ (by AA similarity).
## Step2: Find the ratio of similarity
$ \frac{AE}{AC}=\frac{6}{20}=\frac{3}{10} $.
## Step3: Find $ DE $ and $ AB $
- For $ DE $: $ \frac{DE}{BC}=\frac{3}{10} $, $ DE = 24\times\frac{3}{10}=7.2 \, \text{cm} $.
- For $ AB $: $ \frac{AD}{AB}=\frac{3}{10} $, $ \frac{8}{AB}=\frac{3}{10} $, $ AB=\frac{80}{3}\approx26.67 \, \text{cm} $.
# Answer: $ DE = 7.2 \, \text{cm} $, $ AB=\frac{80}{3} \, \text{cm} $ (or $ \approx26.67 \, \text{cm} $)

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QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Identify similar triangles

Step2: Find the ratio of corresponding sides

Step3: Find \( PQ \)

Step1: Find the scale factor

Step2: Find the remaining sides

Step1: Recall the property of similar figures

Step2: Simplify the ratio

Answer:

Problem 2: