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Area of quadrilateral whose vertices are \( \Large \left(2,\ 3\right),\ \left(3,\ 4\right),\ \left(4,\ 5\right)\ and\ \left(5,\ 6\right) \)

A) 0
B) 4
C) 6
D) none of these

Correct Answer:

A) 0

Description for Correct answer:

The vertices of quadrilateral ABCD is \( \Large A \left(2,\ 3\right),\ B \left(3,\ 4\right),\ C \left(4,\ 5\right)\ and\ D \left(5,\ 6\right) \)

AB = \( \Large \sqrt{ \left(3-2\right)^{2}+ \left(4-3\right)^{2} } = \sqrt{ \left(1\right)^{2}+ \left(1\right)^{2} }= \sqrt{2} \)

Similarly BC = \( \Large \sqrt{2},\ CD=\sqrt{2}\ and\ DA=3\sqrt{2} \)

\( \Large a = b = c = \sqrt{2}\ and\ d = 3\sqrt{2} \)

and \( \Large s=\frac{a+b+c+d}{2}=\frac{\sqrt{2}+\sqrt{2}+\sqrt{2}+3\sqrt{2}}{2} \)

= \( \Large \frac{6\sqrt{2}}{2} = 3\sqrt{2} \)

Therefore, Area of quadrilateral

= \( \Large \sqrt{ \left(s-a\right) \left(s-b\right) \left(s-c\right) \left(s-d\right) } \)

= \( \Large \sqrt{ \left(3\sqrt{2}-\sqrt{2}\right)\left(3\sqrt{2}-\sqrt{2}\right)\left(3\sqrt{2}-\sqrt{2}\right)\left(3\sqrt{2}-3\sqrt{2}\right) } = 0 \).

Part of solved Rectangular and Cartesian products questions and answers : >> Elementary Mathematics >> Rectangular and Cartesian products

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Similar Questions

1). Co-ordinates of the foot of the perpendicular drawn from \( \Large \left(0,\ 0\right) \) to the line joining \( \Large \left(a \cos \alpha ,\ a \sin \alpha\right)\ and\ \left(a \cos \beta ,\ a \sin \beta \right) \) are:

A). \( \Large \left(\frac{a}{2},\ \frac{b}{2}\right) \)

B). \( \Large \left(\frac{a}{2} \left(\cos \alpha + \cos \beta \right), \frac{a}{2} \left(\sin \alpha + \sin \beta \right) \right) \)

C). \( \Large \left(\cos\frac{ \alpha + \beta }{2},\ \sin\frac{ \alpha + \beta }{2}\right) \)

D). \( \Large \left(0,\ \frac{b}{2}\right) \)

2). An equilateral triangle has each side equal to a the co-ordinates of its vertices are \( \Large \left(x_{1},\ y_{1}\right),\ \left(x_{2},\ y_{2}\right)\ and\ \left(x_{3},\ y_{3}\right) \), then the square of determinant \( \begin{vmatrix}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{vmatrix} \) equals

A). \( \Large 3a^{4} \)

B). \( \Large \frac{3}{4}a^{4} \)

C). \( \Large 4a^{4} \)

D). none of these

3). The incentre of triangle with vertices \( \Large \left(1,\ \sqrt{3}\right),\ \left(0,\ 0\right)\ and\ \left(2,\ 0\right) \) is:

A). \( \Large \left(1,\ \frac{\sqrt{3}}{2}\right) \)

B). \( \Large \left(\frac{2}{3},\ \frac{1}{\sqrt{3}}\right) \)

C). \( \Large \left(\frac{2}{3},\ \frac{\sqrt{3}}{2}\right) \)

D). \( \Large \left(1,\ \frac{1}{\sqrt{3}} \right) \)

4). If \( \Large P \left(a_{1},\ b_{1}\right)\ and\ Q \left(a_{2},\ b_{2}\right) \) are two points, the \( \Large OP.OQ \cos \left( \angle POQ\right) \) is (O is origin ):

A). \( \Large a_{1}a_{2}+b_{1}b_{2} \)

B). \( \Large a^{2}_{1}+a^{2}_{2}+b^{2}_{1}+b^{2}_{2} \)

C). \( \Large a^{2}_{1}-a^{2}_{2}+b^{2}_{1}-b^{2}_{2} \)

D). none of these

5). If a point \( \Large P \left(4,\ 3\right) \) is shifted by a distance \( \Large \sqrt{2} \) unit parallel to the line y = x, then co-ordinates of P in new position are:

A). \( \Large \left(5,\ 4\right) \)

B). \( \Large \left(5+\sqrt{2},\ 4+\sqrt{2}\right) \)

C). \( \Large \left(5-\sqrt{2},\ 4-\sqrt{2}\right) \)

D). none of these

6). The orthocentre of a triangle formed by lines \( \Large 2x+y=2,\ x-2y=1\ and\ x+y=1 \) is:

A). \( \Large \left(\frac{1}{3},\ \frac{2}{3}\right) \)

B). \( \Large \left(0,\ 1\right) \)

C). \( \Large \left(\frac{2}{3},\ \frac{1}{3} \right) \)

D). \( \Large \left(1,\ 0\right) \)

7). The incentre of triangle formed by lines x = 0, y = 0 and 3x+4y = 12 is at:

A). \( \Large \left(\frac{1}{2},\ \frac{1}{2}\right) \)

B). \( \Large \left(1,\ 1\right) \)

C). \( \Large \left(1,\ \frac{1}{2} \right) \)

D). \( \Large \left(\frac{1}{2},\ 1\right) \)

8). If the vertices of a triangle have integral co ordinates, the triangle cannot be:

A). an equilateral triangle

B). a right angled triangle

C). an isosceles triangle

D). none of these

9). If \( \Large t_{1}+t_{2}+t_{3}=-t_{1}t_{2}t_{3} \) then orthocentre of the triangle formed by the points \( \Large \left[ at_{1}t_{2},\ a \left(t_{1}+t_{2}\right) \right],\ \left[ at_{2}t_{3},\ a \left(t_{2}+t_{3}\right) \right]\ and\ \left[ at_{3}t_{1},\ a \left(t_{3}-t_{1}\right) \right] \) lies on;

A). \( \Large \left(a,\ 0\right) \)

B). \( \Large \left(-a,\ 0\right) \)

C). \( \Large \left(0,\ a\right) \)

D). \( \Large \left(0,\ -a\right) \)

10). P\( \Large \left(2,\ 1\right) \) is image of the point Q\( \Large \left(4,\ 3\right) \) about the line:

A). x - y = 1

B). 2x - 3y = 0

C). x + y = 5

D). none of these