Question

The vertices of ΔABC are A (1, 5), B (3, 9), and C (5, 3). The vertices of ΔDEF are D (−3, 3), E (−2, 5), and F (−1, 2). Which conclusion is true about the triangles?
A. The ratio of their corresponding sides is 1:3.
B. They are congruent by the definition of congruence in terms of rigid motions.
C. The ratio of their corresponding angles is 1:3.
D. They are similar by the definition of similarity in terms of a dilation.

Answer 1

The dimensions of triangle ΔDEF can be obtained from ΔABC by applying a scale factor of dilation of $\displaystyle \frac{1}{2}$.

• The conclusion which is true about the triangles is D. They are similar by the definition of similarity in terms of dilation.

Reasons:

The given vertices of the triangles are;

ΔABC: A(1, 5), B(3, 9), C(5, 3)

ΔDEF: D(-3, 3), E(-2, 5), and F(-1, 2)

The lengths of the sides of ΔABC are

AB = √((1 - 3)² + (5 - 9)²) = √20 = 2·√5

BC = √((5 - 3)² + (3 - 9)²) = √40 = 2·√10

AC = √((1 - 5)² + (5 - 3)²) = √20 = 2·√10

The lengths of the sides of ΔDEF are;

DE = √((-3 - (-2))² + (3 - 5)²) = √5

EF = FE = √((-1 - (-2))² + (2 - 5)²) = √10

DF = √((-3 - (-1))² + (3 - 2)²) = √5

Therefore;

$\displaystyle The \ dilation \ that \ gives \ \Delta ABC = \frac{\overline {EF}}{\overline {BC}} = \frac{\overline {DE}}{\overline {AB}} = \frac{\overline {DF}}{\overline {AC}} = \frac{\sqrt{10} }{2\cdot \sqrt{10} } = \frac{1}{2}$

The ratio of the sides of ΔABC to the corresponding sides of ΔDEF are equal to $\displaystyle \frac{1}{2}$.

Which gives;

ΔDEF is a dilation of ΔABC, and ΔDEF and ΔABC are similar by the

definition of similarity in terms of a dilation.

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Answer 2

Answer:

Step-by-step explanation:

They are congruent by the definition of congruence in terms of rigid motions.