Distance Formula & Graphing Quiz - ICSE Class 9 Mathematics

Free practice quiz: Distance Formula & Graphing (ICSE Class 9 Mathematics). 42 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 9 Mathematics practice quiz has 42 multiple-choice questions on Distance Formula & Graphing, and 42 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. A line segment PQ has endpoints P(2, 5) and Q(8, 1). What are the coordinates of the midpoint M of PQ?

   1. (4, 3)
   2. (5, 3)
   3. (6, 2)
   4. (5, 4)
   Show answer

   Answer: B. (5, 3)

   The midpoint formula states that for two points (x1, y1) and (x2, y2), the midpoint coordinates are ((x1 + x2)/2, (y1 + y2)/2). Here, P(2, 5) and Q(8, 1). So, the midpoint M is ((2 + 8)/2, (5 + 1)/2) = (10/2, 6/2) = (5, 3).

2. What is the equation of the x-axis?

   1. x = 0
   2. y = 0
   3. x + y = 0
   4. y = x
   Show answer

   Answer: B. y = 0

   The x-axis is the horizontal line where the y-coordinate of every point is zero. Therefore, the equation of the x-axis is y = 0.

3. The distance between points (a, 2) and (3, 6) is 5. Find the possible values of 'a'.

   1. a = 6 or a = 0
   2. a = 5 or a = 1
   3. a = 7 or a = -1
   4. a = 6 or a = -1
   Show answer

   Answer: A. a = 6 or a = 0

   Using the distance formula: sqrt((3 - a)^2 + (6 - 2)^2) = 5. Squaring both sides: (3 - a)^2 + 4^2 = 5^2. (3 - a)^2 + 16 = 25. (3 - a)^2 = 9. So, 3 - a = 3 or 3 - a = -3. If 3 - a = 3, then a = 0. If 3 - a = -3, then a = 3 + 3 = 6. Thus, a = 6 or a = 0.

4. If point P(x, y) is equidistant from A(1, 2) and B(5, 6), what is the equation of the perpendicular bisector of AB?

   1. y = -x + 7
   2. y = x + 3
   3. y = -x + 6
   4. y = x + 4
   Show answer

   Answer: A. y = -x + 7

   First, find the midpoint of AB: M = ((1+5)/2, (2+6)/2) = (6/2, 8/2) = (3, 4). Next, find the slope of AB: m_AB = (6 - 2) / (5 - 1) = 4 / 4 = 1. The slope of the perpendicular bisector (m_perp) is the negative reciprocal of m_AB: m_perp = -1/1 = -1. Now, use the point-slope form with the midpoint M(3, 4) and slope m_perp = -1: y - 4 = -1(x - 3). y - 4 = -x + 3. y = -x + 3 + 4. y = -x + 7.

5. What is the x-coordinate of the point that divides the line segment joining A(-5, 3) and B(4, -7) in the ratio 2:7?

   1. -3
   2. -2
   3. -1
   4. 0
   Show answer

   Answer: A. -3

   Using the section formula for the x-coordinate: x = (m*x2 + n*x1) / (m+n). Here (x1, y1) = (-5, 3), (x2, y2) = (4, -7), m = 2, n = 7. x = (2*4 + 7*(-5)) / (2+7) = (8 - 35) / 9 = -27 / 9 = -3.

More questions in this quiz

Answers and explanations are in the full quiz.

1. Point A is at (-3, 4) and Point B is at (5, -2). Calculate the distance between A and B.
2. Which quadrant does the point (-7, -11) lie in?
3. If the distance between points (k, 0) and (0, 3) is 5 units, find the possible values of k.
4. A point P is on the x-axis such that its distance from point A(2, -3) is 5 units. Find the coordinates of P.
5. What is the slope of the line passing through the points (-2, 3) and (4, -5)?
6. If a line has a slope of 2/3 and passes through the point (3, 4), what is the equation of the line?
7. The vertices of a triangle are O(0, 0), A(a, 0), and B(0, b). What is the area of triangle OAB?
8. If the points (x, y), (1, 2), and (3, 4) are collinear, what is the relationship between x and y?
9. Point P lies on the y-axis such that its distance from A(5, -3) is 13 units. What are the coordinates of P?
10. A point (x, y) is such that its distance from the origin is 5. What is the equation representing its locus?
11. The points P(a, b), Q(c, d), and R(e, f) are vertices of a triangle. What is the area of the triangle using coordinates?
12. Find the coordinates of the point that divides the line segment joining A(-1, 7) and B(4, -3) in the ratio 2:3.
13. What is the y-intercept of the line 3x + 2y = 12?
14. Which of the following points lies on the line y = 2x + 1?
15. What is the x-intercept of the line 4x - 5y = 20?
16. A line passes through (2, 3) and is parallel to the line 3x + 4y = 12. What is the equation of the line?
17. The equation of a line is given by y = mx + c. What does 'm' represent?
18. What is the length of the line segment connecting the points (3, 7) and (3, -2)?
19. If a line passes through the origin and has a slope of -2, what is its equation?
20. The points (0, 0), (x, 0), and (0, y) form a right-angled triangle. Which statement is always true?
21. Two points on a line are (2, 5) and (6, 13). What is the equation of the line?
22. What is the equation of the line perpendicular to y = 2x + 3 and passing through (4, 5)?
23. Which of the following equations represents a vertical line?
24. What is the slope of the line passing through the origin and the point (5, -10)?
25. The line segment joining P(x, y) and Q(3, 8) has a midpoint at M(5, 5). Find the coordinates of P.
26. Which quadrant does the point (8, -3) lie in?
27. What is the equation of the line passing through (0, 4) with a slope of -3?
28. If the points (2, 3), (4, 5), and (x, y) are collinear, what is the relation between x and y?
29. The vertices of a triangle are (0, 0), (8, 0), and (0, 6). What is the length of the hypotenuse?
30. The line segment joining P(x, 2) and Q(6, 10) has its midpoint at M(4, 6). Find the value of x.
31. What is the distance of the point (-3, 5) from the y-axis?
32. The points A(a, b) and B(c, d) are symmetric with respect to the origin. Which of the following must be true?
33. What is the distance of the point (5, -2) from the x-axis?
34. The line segment joining P(x1, y1) and Q(x2, y2) has midpoint M((x1+x2)/2, (y1+y2)/2). If the midpoint of PQ is (2, 3) and P is (5, 7), find Q.
35. Which of the following points lies on the line x = -4?
36. The line x + 2y = 4 is parallel to which of the following lines?
37. What is the distance between the point (0, 0) and the point (a, b)?

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