Question

question 10 of 10
which descriptions from the list below accurately describe the relationship between △abc and △def? check all that apply.
a. same area
b. same size
c. congruent
d. congruent after a dilation

Explanation:

Step1: Calculate the area of \(\triangle ABC\)

The formula for the area of a right - triangle is \(A=\frac{1}{2}\times base\times height\). For \(\triangle ABC\), \(base = 3\) and \(height = 4\). So \(A_{ABC}=\frac{1}{2}\times3\times4=6\)

Step2: Calculate the area of \(\triangle DEF\)

For \(\triangle DEF\), \(base = 6\) and \(height = 8\). So \(A_{DEF}=\frac{1}{2}\times6\times8 = 24\)
Since \(A_{ABC}
eq A_{DEF}\), option A is incorrect.

Step3: Check the side - length ratios

The sides of \(\triangle ABC\) are \(3\), \(4\), \(5\) and the sides of \(\triangle DEF\) are \(6\), \(8\), \(10\). The ratio of corresponding sides is \(\frac{6}{3}=\frac{8}{4}=\frac{10}{5} = 2\)

Step4: Analyze the congruence and similarity concepts

Congruent triangles have equal side - lengths. Since the side - lengths of \(\triangle ABC\) and \(\triangle DEF\) are not equal (\(3
eq6\), \(4
eq8\), \(5
eq10\)), options B and C are incorrect.
A dilation is a transformation that changes the size of a figure but not its shape. If we dilate \(\triangle ABC\) by a scale factor of \(2\), we get \(\triangle DEF\) (because if we multiply each side of \(\triangle ABC\) by \(2\): \(3\times2 = 6\), \(4\times2=8\), \(5\times2 = 10\)). Similar triangles (which can be made congruent by a dilation) have equal corresponding angles (both triangles have angles \(37^{\circ}\), \(53^{\circ}\), \(90^{\circ}\)) and proportional sides.

Answer:

D. Congruent after a dilation