Triangle in Rectangular Cartesian Coordinates

Q. Let $S$ be the set of all triangles in the $xy$-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area $50$ sq. units, then the number of elements in the set $S$ is :

1. $36$
2. $32$
3. $9$
4. $18$

Q. A (3, 4), B (0, 0) and C (3, 0) are vertices of $△ABC.$ If ‘P’ is a point inside $△ABC,$ such that $d(P,BC)\le mind(P,AB),d(P,AC),$ then the maximum of $d(P,BC)$ is $\left(d\right(P,BC)$ represents distance between P. and BC)

1. 
2. 
3. 
4. 

Q.

Find the coordinates of the vertices of a triangle, the equations of whose sides are :

$\left(i\right)x+y-4=0,2x-y+3=0andx-3y+2=0$ $\left(ii\right)y(t_1+t_2)=2x+2a{t}_1{t}_2,y(t_2+t_3)=2x+2a{t}_2{t}_{3}and,y(t_3+t_1)=2x+2a{t}_1{t}_3.$

Q. Let $A(0,1),B(1,1),C(1,-1)$ and $D(-1,0)$ be four points. If $P$ is any other point, then the minimum value of $PA+PB+PC+PD$ is equal to

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

Q. If the vertices A & B of a triangle ABC are given by (2, 5) and (4, –11) respectively. And C moves along the line 9x + 7y + 4=0 then locus of centroid of $\Delta$ ABC is

1. circle
2. parabola
3. parallel to given line
4. perpendicular to given line

Q. Consider a triangle $ABC$ whose one side lies on the line $x+y=4$. If the coordinates of the orthocentre and the centroid are $(0,0)$ and $(2,4)$ respectively, and $R$ is the circumradius, then the value of $3R^2$ is

Q. If the coordinates of the vertices of the triangle ABC be (-1, 6), (- 3, - 9), and (5, -8) respectively, then the equation of the median through C is

1. 13x-14y+47=0
2. 13x-14y-47=0
3. 13x+14y+47=0
4. 13x+14y-47=0

Q. Let $A(h,k),B(1,1)$ and $C(2,1)$ be the vertices of a right angled triangle with $AC$ as its hypotenuse. If the area of the triangle is 1 square unit, then the set of values which ${}^{\prime }{k}^{\prime }$ can take is given by

1. {$-1,3$}
2. {$-3,-2$}
3. {$1,3$}
4. {$0,2$}

Q. If the middle points of the sides BC, CA and AB of the triangle ABC be (1, 3), (5, 7) and (-5, 7), then the equation of the side AB is

1. x-y-2=0
2. x+y-12=0
3. None of these
4. x-y+12=0

Q. If $A(8,12),B(8,0)$ and $C(0,6)$ are the vertices of a triangle $ABC$, then

1. Length of angle bisector through vertex $B$ is $\frac{48}{11}\sqrt{5}$ units
2. Length of angle bisector through vertex $B$ is $\frac{24}{11}\sqrt{5}$ units
3. Incentre divides the angle bisector through $B$ in $11:5$
4. Incentre divides the angle bisector through $B$ in $11:9$

Q. If $P_1$ is the centroid of the triangle $A_1,A_2,A_3,P_2$ is the centroid of the triangle $A_2,A_3,A_4,P_3$ is that of the triangle $A_3,A_4,A_5$ and so on $P_{n-2}$ is the centroid
of the points $A_{n-2},A_{n-1},A_n$. The coordinates of the centre of mean position of $P_1,P_2,...P_{n-2}$ is

1. $(\frac{3(x_1+x_2+...+x_n)}{n-2},\frac{3(y_1+y_2+...+y_n)}{n-2})$
2. $(\frac{x_1+2x_2+3x_3+...+n{x}_n}{n-2},\frac{y_1+2y_2+3y_3+...+n{y}_n}{n-2})$
3. at the centre of mean position of the points
   $A_1,A_2,...A_n$
4. $(\frac{x_1+2x_2+3x_3+...+3x_{n-2}+2x_{n-1}+x_n}{(n-2)},\frac{y_1+2y_2+3y_3+...+3_{y}n-2+2y_{n-1}+y_n}{(n-2)})$

Q. $A(1,0)$ and $B(0,1)$ and two fixed point on the circle $x^2+y^2=1$. C is a variable point on this circle. As C moves, the locus of the orthocentre of the triangle ABC is

1. $x^2+y^2-2x-2y+1=0$
2. $x^2+y^2-x-y=0$
3. $x^2+y^2=4$
4. $x^2+y^2+2x-2y+1=0$

Q. Using the method of integration, find the area of the triangle $ABC$, coordinates of whose vertices are $A(4,1),B(6,6),C(8,4)$

Q. The circum centre of the triangle formed by the points $(2,5,1),(1,4,-3)$ and $(-2,7,-3)$ is

1. $(-1,6,2)$
2. $(6,1,-2)$
3. $(0,6,-1)$
4. $(6,0,1)$

Q.

The vertices of a triangle are (2, 0) (0, 2) (4, 6) then the equation of the median through the vertex (2, 0) is

1. x+y-2=0

2. x=2

3. x+2y-2=0

4. 2x+y-4=0

Q. Let $A(h,k),B(1,1)$ and $C(2,1)$ be the vertices of a right angled triangle with $AC$ as its hypotenuse. If the area of the triangle is 1 square unit, then the set of values which ${}^{\prime }{k}^{\prime }$ can take is given by

1. {$-1,3$}
2. {$-3,-2$}
3. {$1,3$}
4. {$0,2$}

Q. If $A=(-1,6,6)$ , $B=(-4,9,6)$ , $G=\frac{1}{3}(-5,22,22)$ and $G$ is the centroid of the $\Delta ABC$ then the name of the triangle $ABC$ is

1. an isosceles triangle
2. a right angled triangle
3. an equilateral triangle
4. a right-angled isosceles triangle