Question

The points A(4,-2), B(7,2), C(0,9)and D(-3,5) form a parallelogram. Find the length of altitude to the base AB.

Answer 1

Hint: In this question use area of parallelogram $ABCD$ $ = $ area of $\Delta ABC$ + area of $\Delta ACD$. The area of triangle in coordinate form is given as; Area of $\Delta ABC$ = $\dfrac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_{1 - }}{y_2}} \right)} \right]$

Complete step-by-step answer:

First, we will join the diagonal $AC$ and $BD$, we know that the diagonal of a parallelogram bisects each other.

Area of parallelogram $ABCD$$ = $area of $\Delta ABC$ $ + $ area of $\Delta ACD$

Using coordinates
formula, we will find the area of both triangles:
Area of $\Delta ABC$ = $\dfrac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right) + {x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_{1 - }}{y_2}} \right)} \right]$

After putting the values of coordinates which are given in question we get:
$
   = \dfrac{1}{2}\left[ {4\left( {2 - 9} \right) + 7\left( {9 + 2} \right) + 0} \right] \\
   = \dfrac{1}{2}\left[ { - 28 + 77} \right] \\
   = \dfrac{{49}}{2} \\
$

Area of $\Delta ACD$
$
   = \dfrac{1}{2}\left[ {4\left( {9 - 5} \right) + 0\left( {5 + 2} \right) - 3\left( { - 2 - 9} \right)} \right] \\
   = \dfrac{1}{2}\left[ {16 + 33} \right] \\
   = \dfrac{{49}}{2} \\
$

Area of parallelogram $ABCD = \dfrac{{49}}{2} + \dfrac{{49}}{2} = 49$
Now we have to find the length of the altitude from the base $AB$
we know that area of parallelogram is base multiplied by altitude

Base=Distance between coordinates $A$ and $B$, which can be found out by below formula:
$
  \left( {A,B} \right) = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \\
   = \sqrt {{{\left( {2 + 2} \right)}^2} + {{\left( {7 - 4} \right)}^2}} \\
   = \sqrt {{4^2} + {3^2}} \\
   = \sqrt {25} \\
   = 5 \\
$

Area of parallelogram $ABCD = $ Base $ \times $ Altitude $ = 49$
$5 \times $ altitude $ = 49$
Altitude $ = \dfrac{{49}}{5}$

Hence the required value is $\dfrac{{49}}{5}$.

Note: In this question first we found the area of parallelogram $ABCD$ by adding the area of triangles $ABC$ and $ACD$ which are calculated by coordinates formula, after that we found the value of base $AB$ using coordinates formula then we put the values in the formula of parallelogram which is base multiplied by altitude and solved the equation and hence got the length of the altitude to the base $AB$.