Question

directions: list all congruent angles and write a proportion that relates the corresponding sides.

1. △fgh ~ △jkh

directions: the pairs of polygons below are similar. give the scale factor of figure a to figure b.
2.

3.

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

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

Explanation:

1. Congruent angles and proportion for \( \triangle FGH\sim\triangle JKH \)

Step1: Identify congruent angles

• \( \angle F\cong\angle J \) (corresponding angles of similar triangles)
• \( \angle G\cong\angle K \) (corresponding angles of similar triangles)
• \( \angle FHG\cong\angle JHK \) (vertical angles are congruent)

Step2: Write proportion for sides

Since \( \triangle FGH\sim\triangle JKH \), the proportion of corresponding sides is \( \frac{FG}{JK}=\frac{GH}{KH}=\frac{FH}{JH} \)

2. Scale factor for rectangles \( A \) and \( B \)

Step1: Calculate scale factor

The scale factor of figure \( A \) to figure \( B \) is the ratio of corresponding sides. For the sides of length \( 2 \) (in \( A \)) and \( 16 \) (in \( B \)) (or \( 4 \) and \( 8 \)), the scale factor \( k=\frac{2}{16}=\frac{1}{8} \) (or \( \frac{4}{8}=\frac{1}{2} \), but using the pair \( 2 - 16 \) gives \( k = \frac{1}{8} \))

3. Scale factor for triangles \( A \) and \( B \)

Step1: Choose corresponding sides

Take the sides \( 10 \) (in \( A \)) and \( 6 \) (in \( B \)). The scale factor \( k=\frac{10}{6}=\frac{5}{3} \) (or using \( 18 \) and \( 4 \): \( k=\frac{18}{4}=\frac{9}{2} \), but using \( 15 \) and \( 7.2 \): \( k=\frac{15}{7.2}=\frac{150}{72}=\frac{25}{12} \), wait, no, correct way: since \( \triangle A\sim\triangle B \), if we take the side of length \( 10 \) in \( A \) and \( 6 \) in \( B \), scale factor \( k=\frac{10}{6}=\frac{5}{3} \))

4. Find \( x \) when scale factor \( 4:5 \)

Step1: Set up proportion

Since the scale factor of \( A \) to \( B \) is \( \frac{4}{5} \), and the sides are \( x \) (in \( A \)) and \( 15 \) (in \( B \)), we have \( \frac{x}{15}=\frac{4}{5} \)

Step2: Solve for \( x \)

Cross - multiply: \( 5x = 4\times15 \), so \( 5x=60 \), then \( x = 12 \)

5. Find perimeter of \( A \) when scale factor \( 7:2 \)

Step1: Find perimeter of \( B \)

Perimeter of \( B \) is \( P_B=12 + 9+9 + 5=35 \)

Step2: Use scale factor

Since scale factor \( k=\frac{7}{2}=\frac{P_A}{P_B} \), then \( P_A=\frac{7}{2}\times P_B \). Substitute \( P_B = 35 \), \( P_A=\frac{7}{2}\times35=\frac{245}{2}=122.5 \)

Answer:

1.

• Angles: \( \angle F\cong\angle J \), \( \angle G\cong\angle K \), \( \angle FHG\cong\angle JHK \)
• Sides: \( \frac{FG}{JK}=\frac{GH}{KH}=\frac{FH}{JH} \)

1. \( \frac{1}{8} \)
2. \( \frac{5}{3} \)
3. \( 12 \)
4. \( 122.5 \)