Question

quiz #1 a
you must show all of your work to receive credit for a correct answer.
1.) are these two triangles similar?
why or why not?
(images of two triangles with side lengths 30, 50, 40 and 20, 30, 25)
2.) triangle abc is similar to triangle ghr.
what are the pairs of congruent angles?
what is the extended proportion for the ratios of corresponding sides?
3.) what is the scale factor of the following figures?
(images of two right triangles with side lengths 6, 12, 24 and 8, 24, 48)
4.) the lengths of the side of a triangle are in the extended ratio 3:5:7. the perimeter of the triangle is 105. what is the length of the longest side?
(image of a right triangle with side lengths 3x, 5x, 7x)

Explanation:

Step1: Check side ratios (Q1)

Calculate ratios of corresponding sides:
$\frac{30}{20} = \frac{3}{2}$, $\frac{40}{25} = \frac{8}{5}$, $\frac{50}{30} = \frac{5}{3}$

Step2: Compare ratios (Q1)

$\frac{3}{2}
eq \frac{8}{5}
eq \frac{5}{3}$, so sides are not proportional.

Step3: Identify congruent angles (Q2)

For similar $\triangle ABC \sim \triangle GHR$, corresponding angles are congruent:
$\angle A \cong \angle G$, $\angle B \cong \angle H$, $\angle C \cong \angle R$

Step4: Write side proportion (Q2)

Extended proportion of corresponding sides:
$\frac{AB}{GH} = \frac{BC}{HR} = \frac{AC}{GR}$

Step5: Calculate scale factor (Q3)

Take ratio of corresponding sides:
$\frac{6}{8} = \frac{3}{4}$, $\frac{12}{24} = \frac{1}{2}$, $\frac{24}{48} = \frac{1}{2}$. Use consistent ratio: $\frac{\text{small}}{\text{large}} = \frac{1}{2}$

Step6: Set perimeter equation (Q4)

Sum sides equal to perimeter:
$3x + 5x + 7x = 105$

Step7: Solve for x (Q4)

Simplify and solve:
$15x = 105 \implies x = 7$

Step8: Find longest side (Q4)

Longest side is $7x$:
$7 \times 7 = 49$

Answer:

1. No, the side lengths are not in proportional ratios ($\frac{30}{20}

eq \frac{40}{25}
eq \frac{50}{30}$), so the triangles are not similar.

1. Congruent angles: $\angle A \cong \angle G$, $\angle B \cong \angle H$, $\angle C \cong \angle R$; Extended proportion: $\frac{AB}{GH} = \frac{BC}{HR} = \frac{AC}{GR}$
2. Scale factor (small to large) is $\frac{1}{2}$
3. 49