$A (6, 1)$ , $B (8, 2)$ and $C (9, 4)$ are three vertices of a parallelogram $A B C D$ .
If $E$ is the midpoint of $D C$ , find the area of $△ A D E$ .

Open in App

Solution

Step 1: Determine the coordinates of the point $D$
It is given that $A B C D$ is a parallelogram and $A (6, 1)$ , $B (8, 2)$ and $C (9, 4)$ .
Let us assume that $D (h, k)$ .
The midpoint of the diagonal $A C$ is $F \equiv (\frac{6 + 9}{2}, \frac{1 + 4}{2}) = (\frac{15}{2}, \frac{5}{2})$ .
As the diagonals of a parallelogram bisect each other, thus $F$ is also the midpoint of $B D$ .
The midpoint of the diagonal $B D$ can be given by, $F \equiv (\frac{8 + h}{2}, \frac{2 + k}{2})$
$\Rightarrow (\frac{8 + h}{2}, \frac{2 + k}{2}) = (\frac{15}{2}, \frac{5}{2}) \Rightarrow (8 + h, 2 + k) = (15, 5)$
So, $(8 + h) = 15$
$\Rightarrow h = 7$
So, $(2 + k) = 5$
$\Rightarrow k = 3$
Therefore, the coordinate of the point $D$ is $(7, 3)$ .
Step 2: Determine the coordinates of the point $E$
The point $E$ is the midpoint of $C (9, 4)$ and $D (7, 3)$ .
Thus, $E = (\frac{9 + 7}{2}, \frac{4 + 3}{2})$ .
$\Rightarrow E = (\frac{16}{2}, \frac{7}{2}) \Rightarrow E = (8, \frac{7}{2})$
Therefore, the coordinate of the point $E$ is $(8, \frac{7}{2})$ .
Step 3: Determine the area of $△ A D E$
The coordinates of the vertices $A \equiv (6, 1); D \equiv (7, 3); E \equiv (8, \frac{7}{2})$ .
It is known that the area of a triangle with vertices $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})$ can be given by $\frac{1}{2} |x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})|$ .
Thus, the area of $△ A D E = \frac{1}{2} |6 (3 - \frac{7}{2}) + 7 (\frac{7}{2} - 1) + 8 (1 - 3)|$
$= \frac{1}{2} |6 (- \frac{1}{2}) + 7 (\frac{5}{2}) + 8 (- 2)| = \frac{1}{2} |- 3 + \frac{35}{2} - 16| = \frac{1}{2} |\frac{35}{2} - 19| = \frac{1}{2} |\frac{35 - 38}{2}| = \frac{1}{2} |- \frac{3}{2}| = (\frac{1}{2} \times \frac{3}{2}) = \frac{3}{4} {unit}^{2}$
Hence, the area of $△ A D E$ is $\frac{3}{4} {unit}^{2}$ .

Suggest Corrections

Similar questions

Q. A(6, 1), B(8, 2) and C(9, 4) are the vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of

∆ A D E

. [CBSE 2015]

A(6, 1),B(8, 2) and C(9,4) are the vertices of a parallelogram ABCD. If E is the midpoint of BD, find the area of $△ A B E$ .

Let ABCD be a parallelogram of area $124 c m^{2}$ If E and F are the midpoints of sides AB and CD respectively, then find the area of parallelogram AEFD.

Q. A (6, 1), B (8, 2) and C (9, 4) are three vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of

Δ

ADE.

Q. A(6,1) B(8,2) and C(9,4) are three vertices of a parallelogram ABCD. If E is the mid-point of DC, then find the area of

Δ

ADE.

Join BYJU'S Learning Program

A6,1, B8,2 and C9,4 are three vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of △ADE.

$A (6, 1)$ , $B (8, 2)$ and $C (9, 4)$ are three vertices of a parallelogram $A B C D$ .
If $E$ is the midpoint of $D C$ , find the area of $△ A D E$ .