Area of Triangle Using Coordinates

Q. Let the floor of a hall is covered with identical tiles which are in shape of triangle. One of the triangle has the vertices at $(-3,2),(-1,-1)$ and $(1,2)$. If the floor of the hall is completely covered by $110$ tiles, then the area of floor (in sq.units) is

Q.

The number of possible straight lines, passing through $(2,3)$ and forming a triangle with co-ordinate axis, whose area is $12$ sq. units is______.

1. $1$

2. $2$

3. $4$

4. $3$

Q.

ABC is an isosceles triangle. If the coordinates of the base are B (1, 3) and C (- 2, 7), the coordinates of vertex

A can be

1. (1, 6)

2. None of these

3. 

4. 

Q.

Two lines are drawn through $(3,4)$, each of which makes an angle of $45°$ with the line$x-y=2$, then the area of the triangle formed by these lines is

1. $9$

2. $\frac{9}{2}$

3. $2$

4. $\frac{2}{9}$

Q. The coordinates of points $P,Q,R$ and $S$ are $(-3,5),(4,-2),(p,3p)$ and $(6,3)$ respectively, and the area of $△PQR$ and $△QRS$ are in ratio $2:3,$ then possible value of $p$ can be

1. $-\frac{5}{43}$
2. $\frac{3}{43}$
3. $\frac{46}{41}$
4. $\frac{45}{41}$

Q. If the vertices of the $△ABC$ are $A(-3,0),B(4,-1)$ and $C(5,2)$, then the length of the altitude from $A$ to $BC$ is

1. $\frac{15}{\sqrt{7}}\text{units}$
2. $\frac{8}{\sqrt{10}}\text{units}$
3. $\frac{22}{\sqrt{10}}\text{units}$
4. $\frac{22}{\sqrt{7}}\text{units}$

Q.

A (5, 3), B(3, -2) are two fixed points; find the equation to the equation to the locus of a point P which moves so that the area of the triangle PAB is 9 uints.

Q.

Two vertices of a triangle are $(-2,-1)$ and $(3,2)$ and third vertex lies on the line $x+y=5.$ If the area of the triangle is 4 square units, then the third vertex is

1. (0, 5) or, (4, 1)

2. (5, 0) or, (4, 1)

3. (0, 5) or, (1, 4)

4. (5, 0) or, (1, 4)

Q.

The area (in sq. units) of a pentagon whose vertices are $(4,3),(-5,6),(-7,-2),(0,-7)$ and $(3,-6)$ taken in order, is ?

Q. If the mid-points of the sides of a triangle are $(1,1),(2,4)$ and $(3,5)$, then the area of triangle is
(in sq. units)

Q. If the vertices of a triangle are $(1,2),(4,-6)$ and $(3,5)$, then the area (in sq. units) of the triangle is

1. $25$
2. $12$
3. $\frac{25}{2}$
4. $5$

Q.

If A(-1, 1) and B(2, 3) are two fixed points, find the locus of a point P so that the area of $\Delta PAB$ =8sq.units.

Q. Three points $A,B,C$ are taken on an ellipse with eccentric angles $\theta ,\theta +\alpha$ and $\theta +2\alpha$ respectively. The area of $\Delta ABC$ is

1. independent of $\theta$
2. minimum for $\alpha =\frac{2\pi }{3}$
3. maximum for $\alpha =\frac{2\pi }{3}$
4. dependent on $\theta$

Q. If two vertices of an equilateral triangle have integral coordinates, then the coordinates of the third vertex are

1. both irrational
2. both are integers
3. both are rational
4. at least one is irrational

Q.

$A(6,3),B(-3,5),C(4,-2)$ and $D(x,3x)$ are four points. If $\Delta DBC:\Delta ABC=1:2,$ then x is equal to

1. $\frac{11}{8}$

2. $\frac{8}{11}$

3. none of these

4. 3

Q. If a straight line $L$ with negative slope passes through $P(1,4)$ and meets the coordinate axes at $A$ and $B$ such that area of triangle $OAB$ is minimum $(O$ is origin$),$ then which of the following statements is (are) CORRECT ?

1. $y-$intercept of line $L$ is $2.$
2. Distance of line $L$ from $(\sqrt{4},\sqrt{17})$ is $1$ unit
3. Area of triangle $OAB$ is $16$ sq. units
4. Radius of the circle inscribed in triangle $OAB$ is $5-\sqrt{17}$ units

Q.

The area of the circle x² + y² = 16 exterior to the parabola y² = 6x is

A.

B.

C.

D.

Q. The area bounded by y = cos x, y = x + 1, y = 0 is

1. $\frac{3}{2}$
2. $\frac{2}{3}$
3. $\frac{1}{2}$
4. $\frac{5}{2}$

Q. A line $L_1$ passes through the point $(1,1)$ and $(2,0)$ and another line $L_2$ passes through $(\frac{1}{2},0)$ and perpendicular to $L_1.$ Then the area (in sq. units) of the triangle formed by the lines $L_1,$ $L_2$ and $y-$axis is

1. $\frac{25}{4}$
2. $\frac{25}{16}$
3. $\frac{15}{8}$
4. $\frac{25}{8}$

Q. The point $A$ divides the line segment joining the points $(-5,1)$ and $(3,5)$ in the ratio $k:1$. The coordinates of points $B$ and $C$ are $(1,5)$ and $(7,-2)$ respectively. If the area of $△ABC$ is $2$ square units, then the number of values of $k$ is

1. $1$
2. $2$
3. $3$
4. $4$

Q.

The area of a triangle with vertices at $(-4,-1),(1,2)$ and $(4,-3)$ is

1. 17

2. 16

3. 15

4. none of these

Q.

Find area of the triangle with vertices at the point given in each of the following:

(i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8)

(iii) (−2, −3), (3, 2), (−1, −8)

Q. Let $P=(\frac{1}{x_p},p),Q=(\frac{1}{x_q},q)$ and $R=(\frac{1}{x_r},r)$, where $x_k\ne 0$ denotes the $k^{th}$term of an H.P. for $k\in N$. If the area formed by the points $P,Q$ and $R$ is $\lambda pqr$ sq. units, then the value of $\lambda$ is

Q.

Find the area of the triangle formed by the lines

$\left(i\right)y=m_{1}x+c_1,y=m_{2}x+c_{2}andx=0$ $\left(ii\right)y=0,x=2andx+2y=3.$ $\left(iii\right)x+y-6=0,x-3y-2=0and5x-3y+2=0$

Q.

If the vertices of a triangle be $(a{m}_1^2,2a{m}_1),(a{m}_2^2,2a{m}_2)and(a{m}_3^2,2a{m}_3)$, then the area of the triangle is

1. $a(m_2-m_3)(m_3-m_1)(m_1-m_2)$

2. $(m_2-m_3)(m_3-m_1)(m_1-m_2)$

3. $a^2(m_2-m_3)(m_3-m_1)(m_1-m_2)$

4. 4

Q. If the area of the pentagon formed by the vertices $A(1,3),B(-2,5),C(-3,-1),D(0,-2)$ and $E=(2,t)$ is $\frac{45}{2}\text{sq. units}$, then possible value(s) of $t$ is/are

1. $-1$
2. $1$
3. $-99$
4. $89$

Q. The area (in sq. units) of a quadrilateral whose vertices are $A(5,4),$ $B(-3,2),$ $C(-5,-4)$ and $D(7,-6)$ taken in order, is

Q. Let $OPQR$ be a square and $M$ and $N$ be the midpoints of the sides $PQ$ and $QR$ respectively. The ratio of the area of square to the triangle $OMN$ is

1. $4:1$
2. $2:1$
3. $8:3$
4. $7:3$

Q. If the normal of the plane makes angles of $\frac{\pi }{4},\frac{\pi }{4}$ and $\frac{\pi }{2}$ with positive x-axis, y-axis and z-axis respectively and the length of the perpendicular line segment from origin to the plane is $\sqrt{2}$, then the equation of the plane is

1. $x+y+z=\sqrt{2}$
2. $x+y+z=1$
3. $x+y=2$
4. $x=\sqrt{2}$

Q.

If the coordinates of the points A, B, C, be (4, 4), (3, -2) and (3, -16) respectively, then the area of the triangle ABC is

1. 27

2. 15

3. 18

4. 7