Question

1. determine whether the triangles are similar. they may not be drawn to scale.

triangle with sides 8, 10, 14 and another triangle with sides 35, 20, 28 (wait, original second triangle sides: 35, 20, 28? wait no, original image: first triangle (top left) has sides 8, 10, 14; second triangle (below it) has sides 35, 20, 28? wait no, looking back: first triangle (5th problem) has sides 8, 10, 14; second triangle has sides 35, 20, 28? wait no, the second triangle (5th problem) has sides 35, 20, 28? wait the users image: 5. triangle 1: 8,10,14; triangle 2: 35,20,28? wait no, maybe 35,20,28? wait no, the numbers: 8,10,14; then 35,20,28? wait the options: similar or not similar.

1. determine whether the triangles are similar. they may not be drawn to scale.

triangle 1: 3,4,5; triangle 2: 21,28,35. options: similar or not similar.

1. in the diagram below, △abc ~ △def. find n.

△abc: sides 20 (ab), 30 (bc), 45 (ac); △def: sides n (de), 36 (ef), 54 (df).

1. in the diagram below, △abc ~ △def. find x.

△abc: sides 4 (ab), x (bc), 9 (ac); △def: sides 24 (de), 42 (ef), 54 (df).

Explanation:

Question 5

Step1: Order the sides

First triangle sides (sorted): \( 8, 10, 14 \)
Second triangle sides (sorted): \( 20, 25, 35 \)

Step2: Check ratios

Calculate ratios of corresponding sides:
\( \frac{8}{20} = \frac{2}{5} \), \( \frac{10}{25} = \frac{2}{5} \), \( \frac{14}{35} = \frac{2}{5} \)
All ratios are equal.

Step1: Order the sides

First triangle sides (sorted): \( 3, 4, 5 \)
Second triangle sides (sorted): \( 21, 28, 35 \)

Step2: Check ratios

Calculate ratios:
\( \frac{3}{21} = \frac{1}{7} \), \( \frac{4}{28} = \frac{1}{7} \), \( \frac{5}{35} = \frac{1}{7} \)
All ratios are equal.

Step1: Identify corresponding sides

Since \( \triangle ABC \sim \triangle DEF \), corresponding sides are proportional.
Sides of \( \triangle ABC \): \( AB = 20 \), \( BC = 30 \), \( AC = 45 \)
Sides of \( \triangle DEF \): \( DE = n \), \( EF = 36 \), \( DF = 54 \)

Step2: Set up proportion

\( \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \)
Using \( \frac{AB}{DE} = \frac{AC}{DF} \): \( \frac{20}{n} = \frac{45}{54} \)

Step3: Solve for \( n \)

Simplify \( \frac{45}{54} = \frac{5}{6} \)
So \( \frac{20}{n} = \frac{5}{6} \)
Cross - multiply: \( 5n = 20\times6 \)
\( 5n = 120 \)
\( n = \frac{120}{5}=24 \)

Answer:

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