Question

to consider two triangles, first answer part a. then, answer part b.
part a
△abc ≅ △def, ab = 7, bc = 9, ac = 11 + x, df = 3x - 13, and de = 2y - 5
select the figure that represents the congruent triangles.
options as shown in the image (a, b, c, d with triangle diagrams)
part b
find the values of x and y.
x =
y =

Explanation:

Step1: Identify Corresponding Sides

Since \(\triangle ABC \sim \triangle DEF\), their corresponding sides are proportional. From the similar triangles, \(AC\) corresponds to \(DF\) and \(AB\) corresponds to \(DE\). So we set up the equations:
\(AC = DF\) and \(AB = DE\)
Given \(AC = 11 + x\), \(DF = 3x - 13\), \(AB = 7\), \(DE = 2y - 5\).

Step2: Solve for \(x\)

Set \(AC = DF\):
\(11 + x = 3x - 13\)
Subtract \(x\) from both sides:
\(11 = 2x - 13\)
Add 13 to both sides:
\(24 = 2x\)
Divide by 2:
\(x = 12\)

Step3: Solve for \(y\)

Set \(AB = DE\):
\(7 = 2y - 5\)
Add 5 to both sides:
\(12 = 2y\)
Divide by 2:
\(y = 6\)

Answer:

\(x = 12\), \(y = 6\)