Coordinate Geometry · Class 10 Mathematics: 171 MCQs with answers

Q1. The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is

Q2. The distance formula comes from

Q3. The distance of $P(x, y)$ from the origin is

Q4. The distance between $(0, 0)$ and $(3, 4)$ is

Q5. The distance between $(0, 0)$ and $(8, 15)$ is

Q6. The distance between $(2, 3)$ and $(2, 11)$ is

Q7. The distance between $(1, 4)$ and $(7, 4)$ is

Q8. Does the order of the two points change the distance?

Q9. A distance can never be

Q10. Simplified, $\sqrt{50}$ equals

Q11. The distance between $(1, 1)$ and $(9, 7)$ is

Q12. Assertion (A): The distance between $(2, 5)$ and $(5, 9)$ is 5. Reason (R): The horizontal gap is 4 and the vertical gap is 3, and $\sqrt{4^2 + 3^2} = 7$.

Q13. The squared distance between $(2, 1)$ and $(5, 5)$ is

Q14. Before squaring, the two differences between $(4, 9)$ and $(1, 5)$ are

Q15. Two points have the same x-coordinate. The distance between them equals

Q16. Find the distance between the points $(3, 1)$ and $(7, 4)$.

Q17. Find the distance of the point $(-6, 8)$ from the origin.

Q18. The section formula for the ratio $m : n$ from A to B gives the x-coordinate as

Q19. The midpoint of $(x_1, y_1)$ and $(x_2, y_2)$ is

Q20. The midpoint formula is the section formula with

Q21. The midpoint of $(2, 4)$ and $(6, 8)$ is

Q22. The midpoint of $(-3, 5)$ and $(7, -1)$ is

Q23. In the section formula, what must you divide by?

Q24. The two points of trisection divide a segment in the ratios

Q25. A point on the x-axis has

Q26. To find where a segment meets the y-axis you set which expression to zero?

Q27. An unknown ratio is best written as

Q28. Is $3 : 2$ from A the same point as $2 : 3$ from A?

Q29. Assertion (A): The midpoint formula does not need to be memorised separately. Reason (R): It is the section formula with the ratio $1 : 1$.

Q30. The midpoint of $(0, 0)$ and $(10, 6)$ is

Q31. In the ratio $m : n$ from A to B, the part nearer B is

Q32. Assertion (A): The midpoint of a segment is equidistant from its two endpoints. Reason (R): The midpoint divides the segment in the ratio $1 : 1$.

Q33. Find the midpoint of the line segment joining $(4, -3)$ and $(-2, 7)$.

Q34. Three points are collinear when

Q35. To classify a quadrilateral from its vertices you compute

Q36. In a quadrilateral PQRS taken in order, the diagonals are

Q37. A quadrilateral with all four sides equal and both diagonals equal is a

Q38. A quadrilateral with all four sides equal but unequal diagonals is a

Q39. A quadrilateral with opposite sides equal and equal diagonals is a

Q40. Unequal diagonals distinguish a rectangle from a

Q41. Which single check rules out the whole parallelogram family at once?

Q42. A point known to lie on the x-axis should be written as

Q43. For a question about a perimeter, the right tool is

Q44. For a question about trisection, the right tool is

Q45. Assertion (A): Every square is also a rectangle. Reason (R): A rectangle requires opposite sides equal and equal diagonals, both of which a square satisfies.

Q46. Which shape has four equal sides AND four right angles?

Q47. Assertion (A): To find a missing vertex of a parallelogram you equate two midpoints. Reason (R): The diagonals of a parallelogram bisect each other, so they share a midpoint.

Q48. The distance between $(-5, 7)$ and $(-1, 3)$ is

Q49. The distance between $(2, 3)$ and $(4, 1)$ is

Q50. How far is the point $(-3, -4)$ from the origin?

Q51. Comparing $AB^2$ and $AC^2$ instead of $AB$ and $AC$ is valid because

Q52. To show three points form an isosceles triangle, you show that

Q53. Which comparison proves a right angle without any square root?

Q54. For $A(1, 7)$, $B(4, 2)$ and $C(-1, -1)$, the value of $AB^2$ is

Q55. The triangle with vertices $A(1, 7)$, $B(4, 2)$ and $C(-1, -1)$ is

Q56. If the distance between $P(x, 2)$ and $Q(3, 6)$ is 5, then x is

Q57. A general point on the x-axis is written as

Q58. Which coordinate is zero for every point on the y-axis?

Q59. Assertion (A): The distance between $(4, 0)$ and $(0, 3)$ is 5. Reason (R): Both points lie on the axes, so the distance is the sum $4 + 3$.

Q60. Assertion (A): Two points with the same y-coordinate are separated by the difference of their x-coordinates. Reason (R): The vertical gap is then zero, so the formula reduces to $\sqrt{(x_2-x_1)^2} = |x_2 - x_1|$.

Q61. The squared distance between $(0, 5)$ and $(12, 0)$ is

Q62. For $A(0, 0)$, $B(4, 0)$ and $C(0, 3)$, the longest side is

Q63. If $PQ^2 = 45$, then PQ in simplest surd form is

Q64. Find the distance between $(-4, 6)$ and $(2, -2)$.

Q65. Find the distance between $(5, 2)$ and $(3, 4)$, leaving your answer as a surd.

Q66. Show that the points $A(1, 7)$, $B(4, 2)$ and $C(-1, -1)$ are the vertices of an isosceles triangle.

Q67. Find the value of x if the distance between $P(x, 2)$ and $Q(3, 6)$ is 5.

Q68. Explain why comparing squared distances is usually better than comparing the distances themselves.

Q69. Find the perimeter of the triangle with vertices $(0, 0)$, $(4, 0)$ and $(0, 3)$.

Q70. (a) Derive the distance formula from Pythagoras' theorem, explaining each step. (b) Use it to find the distance between $(-3, 5)$ and $(2, -7)$. (c) State two properties of the formula that follow from the differences being squared.

Q71. The point dividing $(-1, 7)$ and $(4, -3)$ in the ratio $2 : 3$ is

Q72. The point dividing $(1, 2)$ and $(7, 8)$ in the ratio $1 : 2$ is

Q73. The first point of trisection of $(4, -1)$ and $(-2, -3)$ is

Q74. The second point of trisection of $(4, -1)$ and $(-2, -3)$ is

Q75. If $P$ divides $A(2, 1)$ and $B(8, 7)$ in the ratio $1 : 2$, then P is

Q76. Setting the y-coordinate of the section formula to zero finds where a segment

Q77. The x-axis divides the join of $(1, -5)$ and $(-4, 5)$ in the ratio

Q78. The point where the x-axis cuts the join of $(1, -5)$ and $(-4, 5)$ is

Q79. If the midpoint of $(a, 4)$ and $(6, b)$ is $(5, 7)$, then a and b are

Q80. Why is an unknown ratio written as $k : 1$?

Q81. After finding a ratio from the x-coordinates, you should

Q82. Assertion (A): The points of trisection divide a segment in the ratios $1 : 3$ and $3 : 1$. Reason (R): Trisection cuts a segment into three equal parts.

Q83. Assertion (A): To find where a segment meets the y-axis, set the section formula's y-coordinate to zero. Reason (R): Every point on the y-axis has x-coordinate zero.

Q84. The point dividing $(1, 1)$ and $(5, 9)$ in the ratio $3 : 1$ is

Q85. A point that is one-quarter of the way from A to B divides AB in the ratio

Q86. If the midpoint of $(2, p)$ and $(q, 8)$ is $(5, 6)$, then p and q are

Q87. Locate the point that splits the segment from $(-1, 7)$ to $(4, -3)$ in the ratio $2 : 3$, and give its coordinates.

Q88. Explain why the section formula has the weights crossed over.

Q89. If the midpoint of the segment joining $(a, 4)$ and $(6, b)$ is $(5, 7)$, find a and b.

Q90. Find the point which divides the join of $(1, 2)$ and $(7, 10)$ in the ratio $3 : 1$.

Q91. Why is it wrong to use the ratio $1 : 3$ for a point of trisection?

Q92. (a) State the section formula and explain what each letter means. (b) Derive the midpoint formula from it. (c) Find the midpoint and the two trisection points of the segment joining $A(2, 1)$ and $B(8, 7)$.

Q93. For collinear points with $AB^2 = 10$, $BC^2 = 40$ and $AC^2 = 90$, the sum $10 + 40$ is

Q94. The points $(-2, 3)$, $(1, 2)$ and $(7, 0)$ are

Q95. The quadrilateral with vertices $(1,1)$, $(5,1)$, $(5,5)$, $(1,5)$ is a

Q96. The quadrilateral with vertices $(0,0)$, $(4,3)$, $(0,6)$, $(-4,3)$ is a

Q97. The quadrilateral with vertices $(0,0)$, $(6,0)$, $(6,3)$, $(0,3)$ is a

Q98. Three vertices of a parallelogram in order are $(1,2)$, $(4,3)$ and $(6,6)$. The fourth is

Q99. Why must the vertices be taken in order?

Q100. If a shape satisfies the conditions for both a rectangle and a rhombus, it is a

Q101. To find a point equidistant from two given points, you

Q102. A question asks what kind of triangle three points form. The right tool is

Q103. A question asks for the fourth vertex of a parallelogram. The right tool is

Q104. Assertion (A): Four equal sides prove a quadrilateral is a square. Reason (R): A rhombus also has four equal sides.

Q105. Assertion (A): Collinearity can be tested using squared distances. Reason (R): Comparing squared distances is valid whenever lengths are being compared for equality.

Q106. The quadrilateral $(4,0)$, $(8,0)$, $(8,4)$, $(4,4)$ is a

Q107. Three collinear points enclose a region of area

Q108. If $AB = 3\sqrt2$, $BC = 2\sqrt2$ and $AC = 5\sqrt2$, the points are

Q109. Writing each distance as a multiple of the same surd helps because

Q110. Show that the points $A(-2, 3)$, $B(1, 2)$ and $C(7, 0)$ are collinear.

Q111. Show that $(1,1)$, $(5,1)$, $(5,5)$ and $(1,5)$ are the vertices of a square.

Q112. Explain why four equal sides do not prove that a quadrilateral is a square.

Q113. State which formula you would use for each: (i) the perimeter of a triangle, (ii) the midpoint of a segment, (iii) whether three points are collinear.

Q114. Why does a question say 'taken in order' when listing the vertices of a quadrilateral?

Q115. The point on the x-axis equidistant from $(2, -5)$ and $(-2, 9)$ is

Q116. Why does setting two squared distances equal give a LINEAR equation?

Q117. The points $(1, 5)$, $(2, 3)$ and $(-2, -11)$ are

Q118. If $(a, 0)$ is equidistant from $(3, 4)$ and $(-1, 2)$, then a is

Q119. A point equidistant from all three vertices of a triangle is

Q120. The triangle with vertices $(0, 0)$, $(4, 0)$ and $(0, 3)$ has perimeter

Q121. If the distance between $(2, -3)$ and $(10, y)$ is 10, then y is

Q122. The points $(3, 0)$, $(6, 4)$ and $(-1, 3)$ form

Q123. Assertion (A): Three points can be tested for a right angle without finding any square root. Reason (R): The converse of Pythagoras can be applied to the squared distances directly.

Q124. Any point at distance 13 from the origin satisfies

Q125. Assertion (A): The distance between $(a, b)$ and $(b, a)$ is zero when $a = b$. Reason (R): The distance between two points is zero exactly when the points coincide.

Q126. Show that the triangle in the previous question is also right-angled, and name the right angle.

Q127. Which point of the x-axis stands the same distance from $(2, -5)$ as it does from $(-2, 9)$?

Q128. Find the values of y for which the distance between $(2, -3)$ and $(10, y)$ is 10.

Q129. The points $A(3, 0)$, $B(6, 4)$ and $C(-1, 3)$ are plotted. (a) Find $AB^2$, $BC^2$ and $CA^2$. (b) Determine what kind of triangle ABC is, justifying every claim. (c) Find the perimeter of the triangle, leaving surds in exact form.

Q130. (a) Find the point on the y-axis which is equidistant from $A(6, 5)$ and $B(-4, 3)$. (b) Explain why setting the two squared distances equal leaves a linear equation. (c) Verify your answer by computing both distances.

Q131. A surveyor plots three markers on a field at $P(0, 0)$, $Q(8, 0)$ and $R(0, 6)$, with all coordinates in metres. (a) Find the distances PQ, PR and QR. (b) Show that the markers form a right-angled triangle and state where the right angle is. (c) A fourth marker S is to be placed so that PQSR is a rectangle. Give its coordinates, and find the distance from P to S.

Q132. The point $(-1, 6)$ divides the join of $(-3, 10)$ and $(6, -8)$ in the ratio

Q133. Taking $(1, 2)$, $(4, y)$, $(x, 6)$ and $(3, 5)$ as the corners of a parallelogram, named in order, x and y come to

Q134. The diagonals of a parallelogram

Q135. If $A(1, 2)$ and $B(7, 10)$, the point three-quarters of the way from A to B divides AB in the ratio

Q136. The point three-quarters of the way from $A(1, 2)$ to $B(7, 10)$ is

Q137. A negative value of k when solving for a ratio means

Q138. The y-axis divides the join of $(-4, -6)$ and $(2, 3)$ in the ratio

Q139. If the midpoint of the join of $(2a, 4)$ and $(-2, 3b)$ is $(1, 5)$, then a and b are

Q140. Assertion (A): Finding a ratio needs only one of the two coordinate equations. Reason (R): The other coordinate provides an independent check on the answer.

Q141. The centroid of a triangle divides each median in the ratio

Q142. The section formula gives a point BETWEEN A and B when

Q143. Split the segment from $(4, -1)$ to $(-2, -3)$ into three equal parts, and name the two points where the cuts fall.

Q144. In what ratio does the point $(-1, 6)$ divide the join of $(-3, 10)$ and $(6, -8)$?

Q145. In what ratio does the x-axis divide the line segment joining $A(1, -5)$ and $B(-4, 5)$? Find the point of division.

Q146. The points $(1, 2)$, $(4, y)$, $(x, 6)$ and $(3, 5)$ are the corners of a parallelogram, named in order. Work out x and y.

Q147. In what ratio does the y-axis divide the join of $(-4, -6)$ and $(2, 3)$?

Q148. (a) Explain the advantage of writing an unknown ratio as $k : 1$. (b) Find the ratio in which $P(-1, 6)$ divides the join of $A(-3, 10)$ and $B(6, -8)$. (c) Verify your answer using the other coordinate.

Q149. (a) Explain how to find the ratio in which the x-axis divides a segment. (b) Find the ratio in which the x-axis divides the join of $A(1, -5)$ and $B(-4, 5)$, and the point of division. (c) Find the ratio in which the y-axis divides the join of $C(-4, -6)$ and $D(2, 3)$.

Q150. During Sports Day on a rectangular ground, flower pots are placed 1 m apart along one edge. Niharika posts a green flag at $(2, 25)$ and Preet posts a red flag at $(8, 20)$, distances in metres. (a) Find the distance between the two flags. (b) Rashmi must post a blue flag exactly halfway between them. Where should she post it? (c) A fourth flag is to go one-third of the way from the green flag to the red flag. Find its position.

Q151. The points $(0, 1)$, $(2, 4)$ and $(5, 7)$ are

Q152. The quadrilateral $(0,0)$, $(5,0)$, $(7,3)$, $(2,3)$ is a

Q153. Three vertices of a rectangle in order are $(0,0)$, $(8,0)$ and $(8,6)$. The fourth is

Q154. A point on the y-axis equidistant from $(6, 5)$ and $(-4, 3)$ is

Q155. For the quadrilateral $(1,1),(5,1),(5,5),(1,5)$, the squared diagonals are

Q156. If three points are collinear, the triangle they 'form' has

Q157. Which pair of checks distinguishes a rectangle from a general parallelogram?

Q158. The fourth vertex D of a parallelogram ABCD with $A(-2,1)$, $B(1,0)$ and $C(4,3)$ is

Q159. Assertion (A): A classification answer should give the most specific shape name. Reason (R): A square also satisfies the definitions of a rectangle, a rhombus and a parallelogram.

Q160. A quadrilateral whose diagonals do NOT have the same midpoint is

Q161. The point on the x-axis equidistant from $(7, 6)$ and $(-3, 4)$ is

Q162. Explain why collinearity must not be tested using squared distances.

Q163. Do $(1, 5)$, $(2, 3)$ and $(-2, -11)$ all lie along one straight line? Justify your answer.

Q164. Show that $(0,0)$, $(4,3)$, $(0,6)$ and $(-4,3)$ form a rhombus but not a square.

Q165. Three vertices of a parallelogram taken in order are $(1, 2)$, $(4, 3)$ and $(6, 6)$. Find the fourth vertex.

Q166. Find the point on the y-axis which is equidistant from $(6, 5)$ and $(-4, 3)$.

Q167. Classify the quadrilateral with vertices $(0,0)$, $(5,0)$, $(7,3)$ and $(2,3)$, justifying your answer.

Q168. (a) State the test for three points to be collinear, and explain why squared distances must not be used. (b) Determine whether $A(3, 1)$, $B(6, 4)$ and $C(8, 6)$ are collinear. (c) Determine whether $P(1, 5)$, $Q(2, 3)$ and $R(-2, -11)$ are collinear.

Q169. (a) List the four classifications of a quadrilateral in terms of its sides and diagonals. (b) Classify $(1,1)$, $(5,1)$, $(5,5)$, $(1,5)$, showing all six squared lengths. (c) Classify $(0,0)$, $(4,3)$, $(0,6)$, $(-4,3)$, and explain what distinguishes it from your answer to (b).

Q170. (a) State the property of a parallelogram used to find a missing vertex. (b) Three vertices of parallelogram ABCD taken in order are $A(-2, 1)$, $B(1, 0)$ and $C(4, 3)$. Find D. (c) Verify your answer by checking that the opposite sides are equal.

Q171. A town planner marks four corners of a proposed park at $P(0, 0)$, $Q(6, 0)$, $R(6, 3)$ and $S(0, 3)$, in metres, taken in order. (a) Find the four squared side lengths and the two squared diagonals. (b) Classify the park, justifying your answer fully. (c) Find the point where the two diagonals cross, and explain how you know they cross there.