Question

If the vertices of $\Delta MNP$ are $M(-4,2),N(-4,6),P(-6,2)$ and of $\Delta QRS$ are $Q(-5,-1),R(-5,-5),S(4,-5)$, then which of the following statements is/are true?

• A. $\Delta MNP$ is congruent to $\Delta QRS$
• B. $\Delta MNP$ is similar to $\Delta QRS$.
• C. $\Delta MNP$ is similar to $\Delta RSQ$.
• D. The triangles are neither congruent nor similar.

Answer 1

Answer: (D)

The triangles are neither congruent nor similar.

Given the vertices of $M(-4,2),N(-4,6),P(-6,2)$, the side lengths are as under:

$MN=\sqrt{(-4+4{)}^2+(2-6{)}^2}=4$,

$MP=\sqrt{(-4+6{)}^2+(2-2{)}^2}=2$,

$NP=\sqrt{(-4+6{)}^2+(6-2{)}^2}=\sqrt{20}$

Similarly, since $Q(-5,-1),R(-5,-5),S(4,-5)$,

$QR=\sqrt{(-5+5{)}^2+(-1+5{)}^2}=4$,

$QS=\sqrt{(-5-4{)}^2+(-1+5{)}^2}=\sqrt{97}$,

$RS=\sqrt{(-5-4{)}^2+(-5+5{)}^2}=9$

Since the sides are not in proportion, in any order and neither are they congruent to each other, the triangles are neither similar nor congruent.