A. Determine the coordinates of the point one-quarter of the way from the point

Question

A. Determine the coordinates of the point one-quarter of the way from the point (a,b) to point (c,d). (Hint: Find the midpoint. How does the quarter-point compare?) B. Determine the coordinates of the point one-eighth of the way from the point (a,b) to point (c,d). C. Guess the coordinates of the point P one-third of the way from (a,b) to (c,d). D. Show that P has the same slope as the line through (a,b) and (c,d). E. Use the distance formula to verify that the distance from (a,b) to P is one-third the distance from (a,b) to (c,d) to see whether you were correct. (If you were not correct, revise your guess in part c.)

Explanation / Answer

PLEASE RATE ME AND AWARD ME KARMA POINTS IF IT IS HELPFUL FOR YOU Coordinates of a point: Each point on a number line is assigned a number. In the same way, each point in a plane is assigned a pair of numbers. x-axis and y-axis: To locate points in a plane, two perpendicular lines are used—a horizontal line called the x-axis and a vertical line called the y-axis. Origin: The point of intersection of the x-axis and y-axis. Coordinate plane: The x-axis, y-axis, and all the points in the plane they determine. Ordered pairs: Every point in a coordinate plane is named by a pair of numbers whose order is important; these numbers are written in parentheses and separated by a comma. x-coordinate : The number to the left of the comma in an ordered pair is the x-coordinate of the point and indicates the amount of movement along the x-axis from the origin. The movement is to the right if the number is positive and to the left if the number is negative. y-coordinate : The number to the right of the comma in an ordered pair is the y-coordinate of the point and indicates the amount of movement perpendicular to the x-axis. The movement is above the x-axis if the number is positive and below the x-axis if the number is negative. Note: The coordinates [ordered pair] for the origin are (0, 0). The x-axis and y-axis separate the coordinate plane into four regions called quadrants. (See Figure 1 .) The upper right quadrant is quadrant 1; the upper left quadrant is quadrant II; the lower left quadrant is quadrant III; and the lower right quadrant is quadrant IV. Notice the following: Figure 1 The coordinate axes separate the plane into four quadrants. In quadrant I, x is always positive and y is always positive. In quadrant II, x is always negative and y is always positive. In quadrant III, x is always negative and y is always negative. In quadrant IV, x is always positive and y is always negative. The point associated with an ordered pair of real numbers is called the graph of the ordered pair.

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