Question

ABCD is quadrilateral, AC = 19cm the length of perpendicular from B and D on AC are 5cm 7cm respectively. Then, the area of ABCD is sq.cm is:

Answer 1

Hint: From the given question, draw a suitable diagram. We will get two triangles separated by the diagonal AC. Then, find the area of the triangle separately and find the area of the quadrilateral by taking the sum of it. To find the area of triangle we will use the formula \[\dfrac{1}{2}\times \text{base}\times \text{height}\]

Complete step-by-step answer:
In the question, we are given a quadrilateral ABCD and we have to find its area from the given data. It is given that, length of AC is 19cm and also length of perpendicular from point B to line AC is 5cm and from point D to line AC is 7cm.
So, according to the question, we will first draw the diagram,

Let the perpendicular from B meet AC point on E and perpendicular from D meet at AC on point F.
So, according to the data given, we can say that BE is 5cm and DF is 7cm.
So, to find the area of the quadrilateral, we will first find areas of triangle ADC and ABC and then add them.
In \[\Delta ABC\] we will find the area using formula \[\dfrac{1}{2}\times \text{base}\times \text{height}\] where base is AC and height is BE. The length of AC is 19cm and BE is 5cm. So, the area will be, \[\dfrac{1}{2}\times 19\times 5\]
Calculation can be written as \[\dfrac{95}{2}\]
In \[\Delta ADC\] we will find the area of triangle by using formula, that is,
\[\dfrac{1}{2}\times \text{base}\times \text{height}\]
Here, the base is AC and height is DF. The length of AC is 19cm and the length of DF is 7cm, then, its area will be,
\[\dfrac{1}{2}\times 19\times 7\]
On calculation we can say that its area is \[\dfrac{133}{2}\]
We know that, the area of \[\Delta ABC\text{ and }\Delta \text{ADC}\] is \[\dfrac{95}{2}c{{m}^{2}}\text{ and }\dfrac{133}{2}c{{m}^{2}}\] respectively. Hence, we can find out the area of quadrilateral ABCD by finding out the sum of area of \[\Delta ABC\text{ and }\Delta \text{ADC}\] which are \[\dfrac{95}{2}c{{m}^{2}}\text{ and }\dfrac{133}{2}c{{m}^{2}}\] respectively. Thus, their sum is \[\left( \dfrac{95}{2}\text{+}\dfrac{133}{2} \right)\Rightarrow \dfrac{228}{2}\Rightarrow 114c{{m}^{2}}\]

Note: We can also find the area of the quadrilateral by another method. If a quadrilateral with a diagonal length is given and if distance of opposite points from the diagonal is given, then, we can directly find the area using formula,
\[\dfrac{1}{2}\times \left( \text{sum of length of distance of opposite points from the diagonal} \right)\times \text{length of diagonal}\]
So, substituting the values, we get the area as,
\[\dfrac{1}{2}\times \left( 5+7 \right)\times 19=\dfrac{1}{2}\times 12\times 19=144c{{m}^{2}}\]