Q1: What is the distance between the points \(A(3, 4)\) and \(B(7, 1)\)? (a) 5 units (b) 25 units (c) 7 units (d) \(\sqrt{7}\) units
Solution:
Ans: (a) Explanation: Using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), we get: \(d = \sqrt{(7-3)^2 + (1-4)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\) units
Q2: What is the slope of a line passing through the points \((2, 5)\) and \((6, 13)\)? (a) 2 (b) 4 (c) \(\frac{1}{2}\) (d) 8
Solution:
Ans: (a) Explanation: Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), we get: \(m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2\)
Q3: The midpoint of the line segment joining points \((2, 8)\) and \((6, 4)\) is: (a) \((4, 6)\) (b) \((8, 12)\) (c) \((2, 2)\) (d) \((3, 5)\)
Solution:
Ans: (a) Explanation: Using the midpoint formula \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\), we get: \(M = \left(\frac{2 + 6}{2}, \frac{8 + 4}{2}\right) = (4, 6)\)
Q4: Which of the following lines is parallel to the line \(y = 3x + 5\)? (a) \(y = 3x - 2\) (b) \(y = -3x + 5\) (c) \(y = \frac{1}{3}x + 1\) (d) \(y = 2x + 5\)
Solution:
Ans: (a) Explanation:Parallel lines have the same slope. The slope of \(y = 3x + 5\) is 3. Option (a) \(y = 3x - 2\) also has a slope of 3, so it is parallel. Option (b) has slope -3, option (c) has slope \(\frac{1}{3}\), and option (d) has slope 2.
Q5: What is the slope of a vertical line? (a) 0 (b) 1 (c) Undefined (d) -1
Solution:
Ans: (c) Explanation: A vertical line has an equation of the form \(x = a\). The slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) gives division by zero, making the slope undefined.
Q6: If two lines are perpendicular and one has a slope of \(\frac{2}{3}\), what is the slope of the other line? (a) \(\frac{2}{3}\) (b) \(-\frac{2}{3}\) (c) \(\frac{3}{2}\) (d) \(-\frac{3}{2}\)
Solution:
Ans: (d) Explanation:Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of \(\frac{2}{3}\) is \(-\frac{3}{2}\).
Q7: What is the equation of a line with slope 4 and y-intercept -3? (a) \(y = 4x + 3\) (b) \(y = -3x + 4\) (c) \(y = 4x - 3\) (d) \(y = 3x - 4\)
Solution:
Ans: (c) Explanation: Using the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, we substitute \(m = 4\) and \(b = -3\) to get \(y = 4x - 3\).
Q8: Point \(P\) divides the line segment joining \(A(1, 3)\) and \(B(5, 11)\) in the ratio 1:1. What are the coordinates of point \(P\)? (a) \((3, 7)\) (b) \((2, 5)\) (c) \((4, 9)\) (d) \((6, 14)\)
Solution:
Ans: (a) Explanation: A point dividing a segment in the ratio 1:1 is the midpoint. Using the midpoint formula: \(P = \left(\frac{1 + 5}{2}, \frac{3 + 11}{2}\right) = (3, 7)\)
## Section B: Fill in the Blanks Q9: The formula to find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is __________.
Solution:
Ans: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) Explanation: This is the distance formula, derived from the Pythagorean theorem, used to calculate the distance between two points in a coordinate plane.
Q10: The slope of a horizontal line is __________.
Solution:
Ans: 0 Explanation: A horizontal line has the form \(y = c\), where all y-values are the same. Since there is no change in y, the slope \(m = \frac{\Delta y}{\Delta x} = \frac{0}{\Delta x} = 0\).
Q11: Two lines are __________ if the product of their slopes is -1.
Solution:
Ans: perpendicular Explanation:Perpendicular lines have slopes that are negative reciprocals, meaning \(m_1 \times m_2 = -1\).
Q12: The point that divides a line segment into two equal parts is called the __________.
Solution:
Ans: midpoint Explanation: The midpoint is the point exactly halfway between two endpoints of a line segment, found using the midpoint formula.
Q13: The standard form of a linear equation is written as __________, where A, B, and C are integers.
Solution:
Ans: \(Ax + By = C\) Explanation: The standard form of a linear equation arranges the variables and constant so that \(Ax + By = C\), where A is typically non-negative.
Q14: If a line passes through the origin, its y-intercept is __________.
Solution:
Ans: 0 Explanation: The y-intercept is the y-coordinate where the line crosses the y-axis. If a line passes through the origin \((0, 0)\), the y-intercept is 0.
## Section C: Word Problems Q15: A map uses a coordinate system where each unit represents 1 kilometer. City A is located at point \((2, 3)\) and City B is at point \((6, 6)\). What is the straight-line distance between the two cities?
Solution:
Ans: Using the distance formula: \(d = \sqrt{(6-2)^2 + (6-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\) Final Answer: 5 kilometers
Q16: A delivery truck travels along a straight road. At 2:00 PM, it is at position \((1, 4)\) on a coordinate grid, and at 3:00 PM, it is at position \((5, 12)\). What is the slope of the truck's path?
Solution:
Ans: Using the slope formula: \(m = \frac{12 - 4}{5 - 1} = \frac{8}{4} = 2\) Final Answer: The slope is 2
Q17: A treasure is buried at the midpoint between two landmarks. One landmark is at coordinates \((-2, 5)\) and the other is at \((4, 9)\). What are the coordinates of the treasure?
Solution:
Ans: Using the midpoint formula: \(M = \left(\frac{-2 + 4}{2}, \frac{5 + 9}{2}\right) = \left(\frac{2}{2}, \frac{14}{2}\right) = (1, 7)\) Final Answer: The treasure is at \((1, 7)\)
Q18: A carpenter is building a ramp. The ramp starts at ground level \((0, 0)\) and ends at point \((12, 3)\), where measurements are in feet. Find the slope of the ramp.
Solution:
Ans: Using the slope formula: \(m = \frac{3 - 0}{12 - 0} = \frac{3}{12} = \frac{1}{4}\) Final Answer: The slope is \(\frac{1}{4}\)
Q19: Two roads intersect perpendicularly. One road has a slope of \(\frac{3}{4}\). Write the equation of the second road if it passes through point \((0, 2)\).
Solution:
Ans: Since the roads are perpendicular, the slope of the second road is the negative reciprocal: \(m = -\frac{4}{3}\) Using point-slope form with point \((0, 2)\): \(y - 2 = -\frac{4}{3}(x - 0)\) Simplifying: \(y = -\frac{4}{3}x + 2\) Final Answer: \(y = -\frac{4}{3}x + 2\)
Q20: Three points \(A(1, 2)\), \(B(4, 6)\), and \(C(7, 10)\) are given. Determine whether these three points are collinear (lie on the same line) by comparing the slopes of segments AB and BC.
Solution:
Ans: Slope of AB: \(m_{AB} = \frac{6 - 2}{4 - 1} = \frac{4}{3}\) Slope of BC: \(m_{BC} = \frac{10 - 6}{7 - 4} = \frac{4}{3}\) Since \(m_{AB} = m_{BC}\), the points are collinear. Final Answer: Yes, the three points are collinear because both slopes equal \(\frac{4}{3}\)
The document Worksheet (with Solutions): Analytic Geometry is a part of the Grade 9 Course Integrated Math 1.
Free, Worksheet (with Solutions): Analytic Geometry, Worksheet (with Solutions): Analytic Geometry, Extra Questions, Viva Questions, Exam, study material, Semester Notes, mock tests for examination, Important questions, Objective type Questions, MCQs, practice quizzes, video lectures, Previous Year Questions with Solutions, Summary, Sample Paper, past year papers, ppt, shortcuts and tricks, Worksheet (with Solutions): Analytic Geometry, pdf ;
