Coordinate Geometry · Class 10 Mathematics: 60 flashcards
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1. State the distance formula.
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$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
2. Which theorem is the distance formula based on?
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Pythagoras' theorem.
3. Does the order of the two points matter?
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No — squaring destroys the sign, so either order gives the same distance.
4. Find the distance between $(1, 1)$ and $(5, 4)$.
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$\sqrt{4^2 + 3^2} = 5$.
5. Give the formula for the distance of $P(x, y)$ from the origin.
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$OP = \sqrt{x^2 + y^2}$
6. Find the distance of $(3, 4)$ from the origin.
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$\sqrt{9 + 16} = 5$.
7. Find the distance between $(-5, 7)$ and $(-1, 3)$.
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$\sqrt{4^2 + (-4)^2} = \sqrt{32} = 4\sqrt2$.
8. Simplify $\sqrt{8}$.
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$2\sqrt2$.
9. Why is comparing squared distances easier?
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They are whole numbers, so no surds or rounding are involved — and $AB^2 = AC^2$ implies $AB = AC$.
10. How do you show three points form an isosceles triangle?
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Show that two of the squared distances are equal.
11. How do you show three points form an equilateral triangle?
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Show all three squared distances are equal.
12. How do you show three points form a right triangle?
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Show the largest squared distance equals the sum of the other two.
13. For $A(1,7)$, $B(4,2)$, $C(-1,-1)$, what are the three squared distances?
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$AB^2 = 34$, $BC^2 = 34$, $CA^2 = 68$.
14. What kind of triangle is $A(1,7)$, $B(4,2)$, $C(-1,-1)$?
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Isosceles AND right-angled at B, since $34 + 34 = 68$.
15. The distance from $P(x, 2)$ to $Q(3, 6)$ is 5. Find x.
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$(3-x)^2 = 9$, so $x = 0$ or $x = 6$.
16. Why do such questions usually have two answers?
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Squaring produces a quadratic, which normally has two roots.
17. How do you write a general point on the x-axis?
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$(x, 0)$.
18. How do you find a point equidistant from two given points?
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Set the two squared distances equal; the $x^2$ terms cancel and a linear equation is left.
19. Can a distance be negative?
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No — always take the positive square root.
20. Should you decimalise $2\sqrt2$?
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Not unless the question asks; leave surds exact and simplified.
21. State the section formula.
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$\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$ for the ratio $m : n$ from A to B.
22. In the section formula, which coordinate does m multiply?
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$x_2$ — the SECOND point's coordinate.
23. Why are the weights crossed?
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A large m puts the point far from A and close to B, so B's coordinates should count for more.
24. State the midpoint formula.
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$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
25. How is the midpoint formula related to the section formula?
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It is the section formula with $m = n = 1$.
26. Find the midpoint of $(1, 2)$ and $(7, 8)$.
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$(4, 5)$.
27. Find the point dividing $(-1, 7)$ and $(4, -3)$ in the ratio $2 : 3$.
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$(1, 3)$.
28. Is $2 : 3$ from A the same point as $3 : 2$ from A?
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No — the first is nearer A, the second nearer B.
29. What ratios give the two points of trisection?
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$1 : 2$ and $2 : 1$.
30. Why is $1 : 3$ wrong for a trisection point?
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It would put the point one part along out of four, not out of three.
31. Find the trisection points of $(4, -1)$ and $(-2, -3)$.
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$\left(2, -\frac53\right)$ and $\left(0, -\frac73\right)$.
32. How should you write an unknown ratio?
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As $k : 1$, which leaves only one unknown.
33. In what ratio does $(-1, 6)$ divide $(-3, 10)$ and $(6, -8)$?
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$2 : 7$, from $\frac{6k-3}{k+1} = -1$.
34. How do you find where a segment crosses the x-axis?
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Set the section formula's y-coordinate to 0 and solve for the ratio.
35. How do you find where a segment crosses the y-axis?
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Set the x-coordinate to 0.
36. In what ratio does the x-axis divide $(1, -5)$ and $(-4, 5)$?
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$1 : 1$ — at the midpoint $\left(-\frac32, 0\right)$.
37. What does a negative value of k mean?
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The point lies outside the segment, on the extension of the line.
38. How do you check a ratio you have found?
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Substitute it back and confirm the other coordinate.
39. What should you write down before substituting?
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Which point is A and which is B.
40. In what ratio does the centroid divide a median?
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$2 : 1$.
41. When are three points collinear?
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When the two shorter distances add to the longest.
42. Why must collinearity use distances, not squared distances?
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Squaring does not respect addition: $(p+q)^2 \ne p^2 + q^2$.
43. Show that $(-2,3)$, $(1,2)$, $(7,0)$ are collinear.
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$\sqrt{10} + 2\sqrt{10} = 3\sqrt{10}$, so $AB + BC = AC$.
44. How many lengths do you compute to classify a quadrilateral?
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Six — four sides and two diagonals.
45. Which vertices do the diagonals join?
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Taken in order, the 1st to the 3rd and the 2nd to the 4th.
46. All four sides equal and diagonals equal — what shape?
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A square.
47. All four sides equal and diagonals unequal — what shape?
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A rhombus.
48. Opposite sides equal and diagonals equal — what shape?
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A rectangle.
49. Opposite sides equal and diagonals unequal — what shape?
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A parallelogram.
50. Why are four equal sides not enough for a square?
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A rhombus also has four equal sides; only the diagonals distinguish them.
51. Classify $(1,1),(5,1),(5,5),(1,5)$.
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A square — all sides 16 and both diagonals 32.
52. Classify $(0,0),(4,3),(0,6),(-4,3)$.
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A rhombus — all sides 25 but diagonals 36 and 64.
53. If a shape is a square, what should you call it?
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A square — the most specific name that applies.
54. Which property finds a missing vertex of a parallelogram?
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The diagonals bisect each other, so both have the same midpoint.
55. Three vertices in order are $(1,2)$, $(4,3)$, $(6,6)$. Find the fourth.
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$(3, 5)$, from equating the midpoints of AC and BD.
56. Why does 'taken in order' matter?
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It tells you which vertices are opposite, and so which pairs form the diagonals.
57. How do you find a point equidistant from two given points?
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Set the two squared distances equal; the $x^2$ terms cancel, leaving a linear equation.
58. How should you write a point known to be on the y-axis?
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As $(0, y)$ — that removes one unknown at the start.
59. Which formula do you use for a ratio or a midpoint?
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The section formula.
60. What must a classification answer contain?
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The six lengths, the comparison, and the most specific name — the comparison carries the marks.
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