Question

For the question given below,four alternative choices have been provided of which only one is correct. You have to select the correct choice:

A triangle and a rhombus are on the same base and between the same parallels. Then, the ratio of area of triangle to that of rhombus is:

(a) 1 : 1

(b) 1 : 2

(c) 1 : 3

(d) 1 : 4

Answer 1

We have to find the ratio of area of triangle to that of rhombus, when they lie on the same base.

Let the triangle be ABC and the rhombus be ABDC.

Therefore, by using the properties of a rhombus that the diagonal divides the rhombus in two equal areas, we get

$$\begin{gathered} \frac{\mathrm{ar}\left(∆\mathrm{ABC}\right)}{\mathrm{ar}\left(\mathrm{ABDC}\right)}=\frac{\mathrm{ar}\left(∆\mathrm{ABC}\right)}{\mathrm{ar}\left(∆\mathrm{ABC}\right)+\mathrm{ar}\left(∆\mathrm{BDC}\right)} \\ =\frac{\mathrm{ar}\left(∆\mathrm{ABC}\right)}{2\mathrm{ar}\left(∆\mathrm{ABC}\right)} \\ =\frac{1}{2} \end{gathered}$$

So, the ratio of the area of triangle to that of rhombus is 1 : 2.

Hence, the correct choice is option (b).