NCERT Exemplar Solutions
Class 10 - Mathematics

CHAPTER 7: Coordinate Geometry

\(\triangle ABC\) with vertices \(A(-2,0),\ B(2,0),\ C(0,2)\) is similar to \(\triangle DEF\) with vertices \(D(-4,0),\ E(4,0),\ F(0,4)\).

Point \(P(-4,2)\) lies on the line segment joining \(A(-4,6)\) and \(B(-4,-6)\).

The points \((0,5),\ (0,-9),\ (3,6)\) are collinear.

\(P(0,2)\) is the intersection of the y–axis and the perpendicular bisector of the segment joining \(A(-1,1)\) and \(B(3,3)\).

Points \(A(3,1),\ B(12,-2),\ C(0,2)\) cannot be vertices of a triangle.

Points \(A(4,3),\ B(6,4),\ C(5,-6),\ D(-3,5)\) are vertices of a parallelogram.

A circle has centre at the origin and a point \(P(5,0)\) lies on it. The point \(Q(6,8)\) lies outside the circle.

The point \(A(2,7)\) lies on the perpendicular bisector of the segment joining \(P(6,5)\) and \(Q(0,-4)\).

The point \(P(5,-3)\) is one of the two trisection points of the segment joining \(A(7,-2)\) and \(B(1,-5)\).

Points \(A(-6,10),\ B(-4,6),\ C(3,-8)\) are collinear such that \(AB=\dfrac{2}{9}\,AC\).

The point \(P(-2,4)\) lies on a circle of radius 6 and centre \((3,5)\).

The points \(A(-1,-2),\ B(4,3),\ C(2,5),\ D(-3,0)\) in that order form a rectangle.

Name the type of triangle formed by the points \(A(-5,6),\ B(-4,-2),\ C(7,5)\).

Find the point(s) on the x–axis at a distance \(2\sqrt5\) from \((7,-4)\). How many such points are there?

What type of quadrilateral is formed by the points \(A(2,-2),\ B(7,3),\ C(11,-1),\ D(6,-6)\) in order?

Find \(a\) if the distance between \(A(-3,-14)\) and \(B(a,-5)\) is \(9\).

Find a point equidistant from \(A(-5,4)\) and \(B(-1,6)\). How many such points are there?

Find the coordinates of the point \(Q\) on the x–axis which lies on the perpendicular bisector of the segment joining \(A(-5,-2)\) and \(B(4,-2)\). Also, name the type of triangle formed by \(Q, A, B\).

Find \(m\) if the points \((5,1),\ (-2,-3),\ (8,2m)\) are collinear.

If \(A(2,-4)\) is equidistant from \(P(3,8)\) and \(Q(-10,y)\), find \(y\). Also find \(PQ\).

Find the area of the triangle with vertices \((-8,4),\ (-6,6),\ (-3,9)\).

In what ratio does the x–axis divide the segment joining \((-4,-6)\) and \((-1,7)\)? Find the coordinates of the point of division.

Find the ratio in which the point \(P\big(\dfrac{3}{4},\dfrac{5}{12}\big)\) divides the segment joining \(A\big(\dfrac{1}{2},\dfrac{3}{2}\big)\) and \(B(2,-5)\).

If \(P(9a-2,-b)\) divides the segment joining \(A(3a+1,-3)\) and \(B(8a,5)\) in the ratio \(3:1\), find \(a\) and \(b\).

If \((a,b)\) is the midpoint of the segment joining \(A(10,-6)\) and \(B(k,4)\) and \(a-2b=18\), find \(k\) and \(|AB|\).

The centre of a circle is \((2a,\,a-7)\). If it passes through \((11,-9)\) and has diameter \(10\sqrt2\), find \(a\).

The line segment joining the points \(A(3,2)\) and \(B(5,1)\) is divided at the point \(P\) in the ratio \(1:2\), and \(P\) lies on the line \(3x - 18y + k = 0\). Find the value of \(k\).

If \(D\big(-\dfrac{1}{2},\dfrac{5}{2}\big),\ E(7,3),\ F\big(\dfrac{7}{2},\dfrac{7}{2}\big)\) are the midpoints of the sides of \(\triangle ABC\), find \(\text{area}(\triangle ABC)\).

The points \(A(2,9),\ B(a,5),\ C(5,5)\) are vertices of a triangle right–angled at \(B\). Find \(a\) and the area of \(\triangle ABC\).

Find the coordinates of the point \(R\) on the segment joining \(P(-1,3)\) and \(Q(2,5)\) such that \(PR=\dfrac{3}{5}PQ\).

Find \(k\) if the points \(A(k+1,2k),\ B(3k,2k+3),\ C(5k-1,5k)\) are collinear.

Find the ratio in which the line \(2x+3y-5=0\) divides the segment joining \(B(8,-9)\) and \(C(2,1)\). Also find the coordinates of the point of division.

If \((-4,\,3)\) and \((4,\,3)\) are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the interior of the triangle.

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

The points \(A(x_1,y_1),\ B(x_2,y_2)\) and \(C(x_3,y_3)\) are the vertices of \(\triangle ABC\).

(i) The median from \(A\) meets \(BC\) at \(D\). Find \(D\).

(ii) Find the point \(P\) on \(AD\) such that \(AP:PD=2:1\).

(iii) Find points \(Q\) and \(R\) on medians \(BE\) and \(CF\) respectively such that \(BQ:QE=2:1\) and \(CR:RF=2:1\).

(iv) Hence, write the coordinates of the centroid of \(\triangle ABC\).

The points \(A(1,-2),\ B(2,3),\ C(a,2)\) and \(D(-4,-3)\) form a parallelogram. Find \(a\) and the height of the parallelogram taking \(AB\) as the base.

Students stand on a grid for drill. Points \(A, B, C, D\) are shown below. Is it possible to place Jaspal so that he is equidistant from all four? If yes, give his position.

(Use grid coordinates: \(A(3,5),\ B(6,8),\ C(10,5),\ D(7,1)\).)

Ayush travels House \(\to\) Bank \(\to\) School \(\to\) Office along straight segments instead of going directly House \(\to\) Office. If \(H(2,4),\ B(5,8),\ S(13,14),\ O(13,26)\), find the extra distance travelled.