CBSE Class 10 Mathematics Coordinate Geometry MCQs Set 08

Question. The centroid of the triangle whose vertices are (3, -7), (-8, 6) and (5, 10) is
(a) (0, 9)
(b) (0, 3)
(c) (1, 3)
(d) (3, 5)
Answer: B
Centroid is \((\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3})\)
i.e. \((\frac{3 - 8 + 5}{3}, \frac{-7 + 6 + 10}{3}) = (\frac{0}{3}, \frac{9}{3}) = (0, 3)\)

Question. The points \(A(-4, -1)\), \(B(-2, -4)\), \(C(4, 0)\) and \(D(2, 3)\) are the vertices of a
(a) Parallelogram
(b) Rectangle
(c) Rhombus
(d) Square
Answer: B

Question. If the point \(P(p, q)\) is equidistant from the points \(A(a+b, b-a)\) and \(B(a-b, a+b)\), then
(a) \(ap = by\)
(b) \(bp = ay\)
(c) \(ap + bq = 0\)
(d) \(bp + aq = 0\)
Answer: D

Question. If the vertices of a triangle have integral coordinates, the triangle cannot be
(a) right angled triangle
(b) isosceles triangle
(c) equilateral triangle
(d) None of the optioms
Answer: C
Let \(A(x_1, y_1)\), \(B(x_2, y_2)\) and \(C(x_3, y_3)\) be the vertices of a \(\Delta ABC\), where \(x_i, y_i, i = 1, 2, 3\) are integers. Then, the area of \(\Delta ABC = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\) is a rational number. If \(\Delta ABC\) is equilateral, its area is \(\frac{\sqrt{3}}{4}(\text{side})^2\), which will be an irrational number, causing a contradiction.

Question. Find the length of the longest side of the triangle formed by the line \(3x + 4y = 12\) with the coordinate axes
(a) 9
(b) 16
(c) 5
(d) 7
Answer: D

Question. Join two points \(P(2, 2)\) and \(Q(4, 2)\) in a plane. Fix the point \(P\) and rotate the line \(PQ\) in anti-clockwise direction at an angle of \(270^\circ\). The area formed by this figure, is
(a) 9 sq units
(b) 9.5 sq units
(c) 9.42 sq units
(d) 9.45 sq units
Answer: C
Radius \(PQ = \sqrt{(4 - 2)^2 + (2 - 2)^2} = 2\) units. Area of the sector with \(270^\circ\) = \(\frac{270}{360} \times \pi \times 2^2 = \frac{3}{4} \times 3.14 \times 4 = 9.42\) sq units.

Question. Suppose there are four points \( A(2, 4), B(6, 4), C(6, 6) \) and \( D(2, 6) \), which lie in the first quadrant. If we rotate only the axes at an angle of \( 90^\circ \) in anti-clockwise direction, then the figure obtained by joining the adjacent points is.
(a) square
(b) rectangle
(c) rhombus
(d) None of the optioms
Answer: B

Question. Area of the region formed by \( 4|x| + 3|y| = 12 \), is
(a) 18 sq units
(b) 20 sq units
(c) 24 sq units
(d) 36 sq units
Answer: C

Question. The circumcentre of the triangle, whose vertices are \( (0, 0), (3, \sqrt{3}) \) and \( (0, 2\sqrt{3}) \), is
(a) \( (1, \sqrt{3}) \)
(b) \( (\sqrt{3}, \sqrt{3}) \)
(c) \( (2, \sqrt{3}, 1) \)
(d) \( (2, \sqrt{3}) \)
Answer: A
\( (1, \sqrt{3}) \)
Let \( O(0, 0), A(3, \sqrt{3}) \) and \( B(0, 2\sqrt{3}) \). Then,
\( OA = \sqrt{3^2 + (\sqrt{3})^2} = \sqrt{12} \)
\( OB = \sqrt{0^2 + (2\sqrt{3})^2} = \sqrt{12} \)
and \( AB = \sqrt{(0-3)^2 + (2\sqrt{3}-\sqrt{3})^2} = \sqrt{9 + (\sqrt{3})^2} = \sqrt{12} \)
\( OA = OB = AB \)
\( \Delta ABC \) is an equilateral triangle.
Now, circumcenter of triangle coincides with centroid of triangle.
Circumcentre of triangle is \( \left( \frac{0+3+0}{3}, \frac{0+\sqrt{3}+2\sqrt{3}}{3} \right) = (1, \sqrt{3}) \)

FILL IN THE BLANK

Question. The point which divide the line segment joining the points \( (5, 4) \) and \( (-6, -7) \) in the ratio \( 1:3 \) internally lies in the .......... quadrant.
Answer: first

Question. Point \( (-4, 6) \) divide the line segment joining the points \( A(-6, 10) \) and \( B(3, -8) \) in the ratio ..........
Answer: \( 2:7 \)

Question. If the coordinates of the points \( P, Q, R \) and \( S \) are such that \( PQ = QR = RS = SP \) and \( PQ \neq QS \), then quadrilateral \( DEFG \) is a ..........
Answer: rhombus

Question. \( (1, 2), (4, y), (x, 6) \) and \( (3, 5) \) are the vertices of a parallelogram taken in order, then the value of \( x \) and \( y \) are ..........
Answer: \( (6, 3) \)

Question. Points \( (1, 5), (2, 3) \) and \( (-2, -11) \) are ..........
Answer: Non-collinear

Question. All the points equidistant from two given points \( A \) and \( B \) lie on the .......... of the line segment \( AB \).
Answer: perpendicular bisector

Question. \( (5, -2), (6, 4) \) and \( (7, -2) \) are the vertices of an .......... triangle.
Answer: isosceles

Question. The distance of a point from the \( y \)-axis is called its ..........
Answer: abscissa

Question. If \( x - y = 2 \) then point \( (x, y) \) is equidistant from \( (7, 1) \) and (..........)
Answer: \( (3, 5) \)

Question. If the co-ordinates of the points \( A, B, C \) and \( D \) are such that \( AB = BC = CD = DA \) and \( AC = BD \), then quadrilateral \( ABCD \) is a ..........
Answer: square

Question. Distance between \( (2, 3) \) and \( (4, 1) \) is ..........
Answer: \( 2\sqrt{2} \)

Question. The distance of a point from the \( x \)-axis is called its ..........
Answer: ordinate

Question. The fourth vertex \( D \) of a parallelogram \( ABCD \) whose three vertices are \( A(-2, 5), B(6, 9) \) and \( C(8, 5) \) is ..........
Answer: \( (0, 1) \)

TRUE/FALSE

Question. The points \( (0, 5), (0, -9) \), and \( (3, 6) \) are collinear.
Answer: False

Question. The distance of the point \( P(3, 2) \) from the \( y \)-axis in 2 units.
Answer: False

Question. The distance of the point \( (5, 3) \) from the \( X \)-axis is 5 units.
Answer: False

Question. Any point on the \( x \)-axis is of the form \( (x, 0) \).
Answer: True

Question. Points \( (1, 7), (4, 2), (-1, -1) \) and \( (-4, 4) \) are the vertices of a square.
Answer: True

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

Question. Coordinates of the point which divides the join of \( (-1, 7) \) and \( (4, -3) \) in the ratio \( 2 : 3 \) is \( (1, 3) \).
Answer: True

Question. The abscissa and ordinate of a point in IV quadrant have same sign.
Answer: False

Question. Ratio in which the line segment joining the points \( (-3, 10) \) and \( (6, -8) \) is divided by \( (-1, 6) \) is \( 3 : 7 \).
Answer: False

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

Question. The distance of a point \( (2, 3) \) from \( Y \)-axis is \( y \)-units.
Answer: False

MATCHING QUESTIONS

DIRECTION : Each question contains statements given in two Columns which have to be matched. Statements (A, B, C, D) in Column-I have to be matched with statements (p, q, r, s) in Column-II.

Question. Column-II gives distance between pair of points given in Column-I.
Column-I
(A) \( (-5, 7), (-1, 3) \)
(B) \( (5, 6), (1, 3) \)
(C) \( (\sqrt{3} + 1, 1), (0, \sqrt{3}) \)
(D) \( (0, 0), (-\sqrt{3}, \sqrt{3}) \)
Column-II
(p) \( \sqrt{17} \)
(q) \( \sqrt{8} \)
(r) \( \sqrt{6} \)
(s) \( 4\sqrt{2} \)
Answer: (A) — s, (B) — p, (C) — q, (D) — r

Question. Column-II gives the coordinates of the point \( P \) that divides the line segment joining the points given in Column-I.
Column-I
(A) \( A(-1, 3) \) and \( B(-5, 6) \) internally in the ratio \( 1 : 2 \)
(B) \( A(-2, 1) \) and \( B(1, 4) \) internally in the ratio \( 2 : 1 \)
(C) \( A(-1, 7) \) and \( B(4, -3) \) internally in the ratio \( 2 : 3 \)
(D) \( A(4, -3) \) and \( B(8, 5) \) internally in the ratio \( 3 : 1 \)
Column-II
(p) \( (7, 3) \)
(q) \( (0, 3) \)
(r) \( (1, 3) \)
(s) \( (1, 0) \)
Answer: (A) — s, (B) — q, (C) — r, (D) — p

Question. Column-II gives the area of triangles whose vertices are given in Column-I.
Column-I
(A) \( (2, 3), (-1, 0), (2, -4) \)
(B) \( (-5, -1), (3, -5), (5, 2) \)
(C) \( (1, -1), (-4, 6), (-3, -5) \)
(D) \( (0, 0), (8, 0), (0, 10) \)
Column-II
(p) 40
(q) 24
(r) 32
(s) 10.5
Answer: (A) — s, (B) — r, (C) — q, (D) — p

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : Mid-point of a line segment divides line in the ratio \( 1 : 1 \).
Reason : If area of triangle is zero that means points are collinear.
Answer: B
Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A). Both statements are individually correct.

Question. Assertion : Centroid of a triangle formed by the points \( (a, b), (b, c) \) and \( (c, a) \) is at origin, Then \( a+b+c = 0 \).
Reason : Centroid of a \( \triangle ABC \) with vertices \( A(x_1, y_1), B(x_2, y_2) \) and \( C(x_3, y_3) \) is given by \( \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \).
Answer: A
Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A). Centroid is \( \left( \frac{a+b+c}{3}, \frac{b+c+a}{3} \right) = (0, 0) \Rightarrow a+b+c = 0 \).

Question. Assertion : The points \( (k, 2-2k), (-k+1, 2k) \) and \( (-4-k, 6-2k) \) are collinear if \( k = \frac{1}{2} \).
Reason : Three points \( A, B \) and \( C \) are collinear in same straight line, if \( AB + BC = AC \).
Answer: A
Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion : The area of the triangle with vertices \( (-5, -1), (3, -5), (5, 2) \), is 32 square units.
Reason : The point \( (x, y) \) divides the line segment joining the points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( k : 1 \) externally then \( x = \frac{kx_2 + x_1}{k + 1}, y = \frac{ky_2 + y_1}{k + 1} \).
Answer: C
Assertion (A) is true but reason (R) is false. Area of triangle \( = \frac{1}{2} |-5(-5 - 2) + 3(2 + 1) + 5(-1 + 5)| = \frac{1}{2} [35 + 9 + 20] = \frac{1}{2} \times 64 = 32 \). External section formula uses minus sign.