Question

\[\vartriangle ABC\] is similar to \[\vartriangle PQR\]. \[AB\] corresponds to \[PQ\] and \[BC\] corresponds to \[QR\]. If \[AB = 9\], \[BC = 12\], \[CA = 6\] and \[PQ = 3\], what are the lengths of \[QR\] and \[RP\]?

Answer 1

Hint: Given that \[\vartriangle ABC\] is similar to \[\vartriangle PQR\]. As we know that if two triangles are similar then corresponding sides will be in the same proportions. So, we will get the relation \[\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}} = \dfrac{{AC}}{{PR}}\]. Then we will put the given values i.e., \[AB = 9\], \[BC = 12\], \[CA = 6\] and \[PQ = 3\]. We will consider the first and the second term and then the first and the third term to find \[QR\] and \[RP\] respectively.

Complete step by step answer:
Given, two triangles \[\vartriangle ABC\] and \[\vartriangle PQR\] are similar.

As we know that if two triangles are similar then corresponding angles will be the same and corresponding sides will be in the same proportions. So, we get \[\angle A = \angle P\], \[\angle B = \angle Q\], \[\angle C = \angle R\] and \[\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}} = \dfrac{{AC}}{{PR}}\].
Here, given that \[AB = 9\], \[BC = 12\], \[CA = 6\] and \[PQ = 3\].
Putting all these values in the ratio of sides, we get
\[ \Rightarrow \dfrac{9}{3} = \dfrac{{12}}{{QR}} = \dfrac{6}{{PR}}\]
Considering first two terms, we get
\[ \Rightarrow \dfrac{9}{3} = \dfrac{{12}}{{QR}}\]
On cross multiplication, we get
\[ \Rightarrow QR = \dfrac{{12 \times 3}}{9}\]
On simplification, we get
\[ \Rightarrow QR = 4\]
Considering first and third terms, we get
\[ \Rightarrow \dfrac{9}{3} = \dfrac{6}{{PR}}\]
On cross multiplication, we get
\[ \Rightarrow PR = \dfrac{{6 \times 3}}{9}\]
\[ \Rightarrow PR = 2\]
Therefore, the length of \[QR\] is \[4{\text{ units}}\]and \[RP\] is \[2{\text{ units}}\].

Note:
Triangles are said to be similar if they have the same shape but it is not necessary that they have the same size. To find if two triangles are similar, there are three ways. AA (angle-angle) means two corresponding angles are equal in two triangles. SAS (side-angle-side) means that two triangles have two sides proportional and the corresponding angle is the same. SSS (side-side-side) means that the corresponding sides of two triangles are in the same ratio.