Question

PQRS is a parallelogram. QM is the height from Q to SR and QN is the height from Q to PS. If SR = 12cm, QM = 7.6cm. Find QN if PS = 8cm.

Answer 1

Hint: Use the fact that the area of a parallelogram is equal to $base\times height$. Hence find the area of the parallelogram in two different ways. Equate the two areas and hence form an equation in QN. Solve the equation and hence find the value of QN.

Complete step-by-step answer:

We know that area of a parallelogram $=base\times height$
With QM as altitude and SR as base, we have
The area of the parallelogram PQRS $=QM\times SR$
Given that QM = 12cm and SR =7.6cm, we have
The area of the parallelogram PQRS $=12\times 7.6=91.2$square centimetres
Also, taking QN as altitude and PS as the corresponding base, we have
The area of the parallelogram PQRS $=QN\times PS$
Given that PS =8cm, we have
The area of the parallelogram PQRS = 8QN square centimetres.
Since the area of the parallelogram is unique, we have
8QN = 91.2
Dividing both sides by 8, we get
$QN=\dfrac{91.2}{8}=11.4$
Hence the length of QN = 11.4 cm.

Note: In the questions of the above type we can use ratio and proportion also to find the relation between the altitude and the base.
We have $area=base\times height\Rightarrow base\propto \dfrac{1}{height}$
Hence, we have
$\dfrac{{{h}_{1}}}{{{h}_{2}}}=\dfrac{{{b}_{2}}}{{{b}_{1}}}$
Here, we have ${{h}_{1}}=QN,{{h}_{2}}=QM,{{b}_{1}}=PS$ and ${{b}_{2}}=SR.$
Hence, we have
$\dfrac{QN}{QM}=\dfrac{SR}{PS}$
Substituting QM = 12, SR = 7.6 and PS = 8, we get
$\dfrac{QN}{12}=\dfrac{7.6}{8}\Rightarrow QN=11.6$, which is the same as obtained above.