Question

Wich of the following is true if ABC~ QPR

Answer 1

We are given that the triangles ABC and QPR are similar.

Recall the properties of similar triangles,

• Corresponding angles are equal.

,

• Corresponding sides are in the same ratio.

Let us first identify the corresponding sides

Side AB = Side QP

Side BC = Side RP

Side AC = Side QR

Now let us find the missing side length RP using the ratio of corresponding sides.

`\begin{gathered} (QP)/(AB)=(RP)/(BC) \\ (5)/(10)=(RP)/(9) \\ RP=(5)/(10)\cdot9 \\ RP=4.5 \end{gathered}`

So, the length of the side RP is 4.5 hence the 2nd option is correct.

Now let us check the corresponding angles.

m∠R = m∠C

m∠Q = m∠A

m∠P = m∠B

From the figure, we see that

`\begin{gathered} m\angle R=m\angle C=81\degree \\ m\angle Q=m\angle A=63\degree \end{gathered}`

Hence the 1st option and the 3rd option are correct.

Recall that the sum of the interior angles of a triangle is equal to 180°

`\begin{gathered} m\angle R+m\angle Q+m\angle P=180\degree \\ 81\degree+63\degree+m\angle P=180\degree \\ 144\degree+m\angle P=180\degree \\ m\angle P=180\degree-144\degree \\ m\angle P=36\degree \end{gathered}`

Hence, the last option is incorrect since the m∠P is 36° (not 81°)

Therefore, the correct options are 1st, 2nd, and 3rd only.