Explore the essential concepts of coordinate geometry with beginner-level questions covering points, lines, slopes, distances, and midpoints on the Cartesian plane. Strengthen your understanding of key terms and simple calculations using coordinates, designed for those starting their journey in analytic geometry.
What are the coordinates of the point located 3 units to the right and 2 units up from the origin on a Cartesian plane?
Explanation: The point 3 units right and 2 units up from the origin has coordinates (3, 2), where 3 is the x-coordinate and 2 is the y-coordinate. Option (2, 3) reverses the order of x and y, which is incorrect. Option (-3, 2) reflects across the y-axis and (3, -2) reflects across the x-axis, both placing the point in wrong positions. The order and the signs are crucial to define a point's correct location.
Which axis does the point (0, 5) lie on in the coordinate plane?
Explanation: The point (0, 5) lies on the y-axis because its x-coordinate is 0 and the y-coordinate is not zero, indicating it is positioned along the vertical axis. The x-axis is where y equals zero, so (0, 5) does not lie there. The origin is the point (0, 0), and (0, 5) is not at the origin. Quadrant II contains points with negative x and positive y, but on an axis is not considered inside any quadrant.
What is the slope of the line passing through the points (2, 3) and (4, 7)?
Explanation: The slope is calculated as the change in y divided by the change in x (m = (7-3)/(4-2) = 4/2 = 2). Option 3 results from using the wrong values, and option 4 results from dividing 4 by 1 instead of by 2. Option 1 ignores the actual differences in values. Correct substitution is vital to find the correct slope.
What is the midpoint between the points (2, 6) and (4, 2)?
Explanation: The midpoint is calculated with the formula ((x1 + x2)/2, (y1 + y2)/2), so ((2+4)/2, (6+2)/2) = (3, 4). Option (2, 4) averages only one coordinate correctly, while (4, 6) simply repeats the original endpoints. Option (3, 2) uses the correct x but incorrect y. Precision in averaging both coordinates gives the correct answer.
In which quadrant is the point (-7, 4) located?
Explanation: Quadrant II contains points where x is negative and y is positive, as with (-7, 4). Quadrant I has both coordinates positive, Quadrant III has both negative, and Quadrant IV has x positive and y negative. Recognizing the signs helps in quickly identifying the correct quadrant.
What is the distance between the points (1, 2) and (4, 6)?
Explanation: The distance formula gives √[(4-1)² + (6-2)²] = √[9+16] = √25 = 5. Option 7 confuses addition of differences, not their squares. Option 4 reflects only the x difference, and option 3 only the y difference. The combination of squared differences is necessary for accurate distance calculation.
What is the equation of a line with slope 3 passing through the origin?
Explanation: A line with slope m through the origin (0, 0) has the equation y = mx, so here y = 3x. Option 'y = x + 3' has the correct form but wrong slope and intercept, 'y = 3' represents a horizontal line, and 'y = x - 3' is a different slope and intercept. Both slope and intercept must match the given conditions.
Which equation represents a vertical line on the coordinate plane?
Explanation: Vertical lines have equations of the form x = constant, such as x = 5. 'y = 5' is a horizontal line. 'y = x' is a diagonal, and 'y = 2x + 1' is also a slanted line. The defining feature of a vertical line is that all points have the same x-coordinate.
What are the coordinates of the origin on the Cartesian plane?
Explanation: The origin is the point where the x-axis and y-axis intersect, which is at (0, 0). The other options provide coordinates that are each one unit away from the origin along either axis or both, but do not represent the intersection point. The origin is always at zero for both coordinates.
In the ordered pair (x, y), what does the 'x' value represent?
Explanation: The x value in an ordered pair (x, y) shows the horizontal position on the plane. The vertical position is given by the y value, not x. 'Origin' refers to the point (0, 0), not a value within the pair, and 'distance from origin' is not the meaning of x but a separate calculation. Understanding the role of x helps in reading and plotting points correctly.