Question

\[\Delta ABC ~ \Delta PQR\], the ratio of their corresponding altitudes AD and PS are in\[2:3\], then \[\Delta ABC:\Delta PQR = \]…………
A. \[9:4\]
B. \[16:81\]
C. \[4:9\]
D. \[2:3\]

Answer 1

Hint: We know that the sum of the angles of a triangle is taken as \[180^\circ \]. We can say that two triangles are similar triangles only when the two triangles are equal in shape and different in sizes or it can be said that the two triangles are similar to each other whenever they are in the same proportion.


Complete step by step solution
Given:
The ratio of the altitudes of AD and PS is\[x:y = 2:3\].
The following are the schematic diagram of the triangles ABC and PQR.

Given that \[\Delta ABC\]and\[\Delta PQR\] are similar to each other. Here we can observe that the two triangles are similar but ratios are different, it means these triangles are similar according to their proportion and shape.
The ratios of the two triangles can be found by the ratios of the squares of the two altitudes that are given. It can be written as:
\[\dfrac{{\Delta ABC}}{{\Delta PQR}} = \dfrac{{{x^2}}}{{{y^2}}}\]
On substituting the values of x and y in the above equation, we get,
\[\begin{array}{l}
\dfrac{{\Delta ABC}}{{\Delta PQR}} = \dfrac{{{x^2}}}{{{y^2}}}\\
\dfrac{{\Delta ABC}}{{\Delta PQR}} = \dfrac{{{2^2}}}{{{3^2}}}\\
\dfrac{{\Delta ABC}}{{\Delta PQR}} = \dfrac{4}{9}
\end{array}\]
Therefore, it is obtained that the ratio of \[\Delta ABC\] and \[\Delta PQR\] is \[4:9\], which means the option (c) is correct.


Note: Here, in this problem we have to be careful while taking the ratios, the question is asked as the ratio of triangle ABC and PQR, so the altitude of the triangle ABC should be in the numerator and the altitude of the triangle PQR should be in the denominator. The ratios of the \[\Delta ABC\] and \[\Delta PQR\] can also be written as the ratios of the area of the triangle \[\Delta ABC\] and the area of the triangle \[\Delta PQR\].