Question

In a parallelogram, ABCD, if AD = 10cm, CF = 8cm then find AB.

Answer 1

Hint: Area of parallelogram is given by the product of any side of it and corresponding height (perpendicular) of it from another opposite side of it. Use this formula for calculation area of parallelogram in given two conditions. Form two equations of writing area and equate them to get the side AB.

Complete step-by-step answer:
As we know area of any parallelogram can be given as

$\text{Area of parallelogram = Side }\!\!\times\!\!\text{ Corresponding height}$ ………… (i)

Diagram of parallelogram ABCD is given as

Where CF is perpendicular to line segment AB and BE is perpendicular to AD.

 Now, as we know, the area of any parallelogram can be calculated with the help of any side of it and its corresponding height from the equation (i).

As side AB and CF are perpendicular to each other, it means we can write the area of parallelogram by using the equation (i) as

$\text{Area of ABCD = AB }\!\!\times\!\!\text{ CF}$

We are given that side CF is of length 8cm.

Hence, we get area of ABCD as

$\text{Area of ABCD = 8 }\!\!\times\!\!\text{ ABc}{{\text{m}}^{\text{2}}}$ ………….. (ii)

Now, we can calculate the area of the same parallelogram ABCD using the relation (i), with the help of side AD and perpendicular BE. So, we know

AD = 10cm, BE = 12cm

$\begin{align}

  & \text{Area of ABCD = 10 }\times \text{ 12} \\

 & =120c{{m}^{2}} \\

\end{align}$ ……………. (iii)

Now, we know that the area of ABCD will be the same, either we calculate by equation (ii) or (iii). So, we can equate area of ABCD from equation (ii) and (iii) and hence, we get

$\begin{align}

  & \text{Area of ABCD = 8}\times \text{AB = 120} \\

 & 8\times AB=120 \\

\end{align}$

Divide the whole equation by 8, we get

$\dfrac{8\times AB}{8}=\dfrac{120}{8}=15$

AB = 15cm

Hence, the length of side AB is 15cm.

Note: Be careful with the side and corresponding height in the formula of area of parallelogram. Don’t multiply the side with any other perpendicular which is not making ${{90}^{\circ }}$with the side. It is the key point of the problem.
One may confuse the area of the rectangle and may use the formula of area as $length\times breadth$, which is wrong. So, be clear with the formulae of both and don’t confuse yourself with any other formula.