Question

which is true about triangles abc and def?
graph with triangles abc and def on a coordinate grid

• the triangles are similar because \\(\frac{12 - 6}{4 - 2} = \frac{27 - 15}{9 - 5}\\)
• the triangles are congruent because \\(\frac{12 - 6}{4 - 2} = \frac{27 - 15}{9 - 5}\\)
• the triangles are similar because \\(\frac{6 - 12}{4 - 2} = \frac{27 - 15}{9 - 5}\\)
• the triangles are congruent because \\(\frac{12 - 6}{4 - 2} = \frac{15 - 27}{9 - 5}\\)

Explanation:

Step1: Identify triangle coordinates

Triangle ABC: $A(2,6)$, $B(4,6)$, $C(4,12)$
Triangle DEF: $D(5,15)$, $E(9,15)$, $F(9,27)$

Step2: Calculate side lengths

For ABC:
$AB = 4-2 = 2$, $BC = 12-6 = 6$
For DEF:
$DE = 9-5 = 4$, $EF = 27-15 = 12$

Step3: Check similarity ratio

Calculate ratios of corresponding sides:
$\frac{BC}{AB} = \frac{12-6}{4-2} = \frac{6}{2} = 3$
$\frac{EF}{DE} = \frac{27-15}{9-5} = \frac{12}{4} = 3$
Since ratios are equal, triangles are similar (not congruent, as sides are not equal).

Step4: Match to correct option

The valid ratio equation matches the first option.

Answer:

The triangles are similar because $\frac{12-6}{4-2} = \frac{27-15}{9-5}$