Question

1. solve for x. 4(2x - 4) = 2(x - 5)

\frac{2x - 4}{2} = \frac{x - 5}{4}

1. qrs ~ nop. list the pairs of congruent angles. write the extended proportion that relates the corresponding sides for the similar triangles.
2. the pair of polygons shown below is similar. what is the scale factor from δxzy to δabc?
3. are the two triangles similar? how do you know?

Explanation:

Problem 33: Solve for \( x \) in \( 4(2x - 4) = 2(x - 5) \)

Step 1: Distribute both sides

Distribute the 4 on the left and 2 on the right:
\( 8x - 16 = 2x - 10 \)

Step 2: Subtract \( 2x \) from both sides

\( 8x - 2x - 16 = -10 \)
\( 6x - 16 = -10 \)

Step 3: Add 16 to both sides

\( 6x = -10 + 16 \)
\( 6x = 6 \)

Step 4: Divide by 6

\( x = \frac{6}{6} = 1 \)

• Congruent Angles: In similar triangles, corresponding angles are congruent. So \( \angle Q \cong \angle N \), \( \angle R \cong \angle O \), \( \angle S \cong \angle P \).
• Extended Proportion: Corresponding sides are proportional. So \( \frac{QR}{NO} = \frac{RS}{OP} = \frac{QS}{NP} \).

Step 1: Identify corresponding sides

• \( XY = 8 \), \( AB = 4 \)
• \( XZ = 6 \), \( AC = 3 \)
• \( YZ = 10 \), \( BC = 5 \)

Step 2: Calculate the ratio of corresponding sides

Scale factor = \( \frac{\text{Side of } \triangle ABC}{\text{Side of } \triangle XZY} \) (since we go from \( \triangle XZY \) to \( \triangle ABC \), the smaller triangle).
For \( XY \) and \( AB \): \( \frac{4}{8} = \frac{1}{2} \)
For \( XZ \) and \( AC \): \( \frac{3}{6} = \frac{1}{2} \)
For \( YZ \) and \( BC \): \( \frac{5}{10} = \frac{1}{2} \)

Answer:

\( x = 1 \)

Problem 35: Similar Triangles \( \triangle QRS \sim \triangle NOP \)