Math Problem Statement

YAY

Determine the coordinates of point C that divides AB into two segments in the ratio 1:3.

Solution

The question in the image asks to determine the coordinates of point CC that divides the line segment ABAB in the ratio 1:31:3. The coordinates of AA and BB are given, and intermediate steps show how to compute the slope, the run, and the rise.

Here's a breakdown of the steps involved:

  1. Coordinates of points:

    • Point AA is located at (0,0)(0, 0).
    • Point BB is located at (8,3)(-8, 3).
  2. Slope of the line: The slope of the line is calculated using the formula: slope=y2y1x2x1=3080=38=0.375.\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{-8 - 0} = \frac{3}{-8} = -0.375. This value is correctly computed in the image.

  3. Ratio and division of the segment: The ratio of the division is 1:31:3, meaning that the point CC divides the segment into two parts where one part is 14\frac{1}{4} (since 11+3\frac{1}{1+3}) of the total length.

  4. Run and rise:

    • The total run (horizontal change) from BB to AA is 88 units (from 8-8 to 00).
    • Taking 14\frac{1}{4} of the run: 14×8=2\frac{1}{4} \times 8 = 2. Thus, the x-coordinate of CC is 8+2=6-8 + 2 = -6.
    • The total rise (vertical change) is 33 units.
    • Taking 14\frac{1}{4} of the rise: 14×3=0.75\frac{1}{4} \times 3 = 0.75. So, the y-coordinate of CC is 0+0.75=0.750 + 0.75 = 0.75.
  5. Coordinates of point CC: Thus, point CC is located at (6,0.75)(-6, 0.75).

Solution Summary:

  • Point CC, which divides ABAB in the ratio 1:31:3, is located at (6,0.75)(-6, 0.75).

Would you like more details on any step or a deeper explanation?

Here are five related questions to explore:

  1. How is the concept of slope useful in determining points along a line?
  2. What are other methods to divide a line segment into a given ratio?
  3. Can you apply the midpoint formula to verify the division of the segment?
  4. How do you generalize this to find a point dividing any two points in a ratio m:nm:n?
  5. What would the coordinates be if the ratio was reversed, say 3:13:1?

Tip: The formula to divide a line segment in the ratio m:nm:n between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is: (mx2+nx1m+n,my2+ny1m+n).\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right).

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Line Division
Ratios

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Division of line segment in a ratio m:n: (mx2 + nx1) / (m+n), (my2 + ny1) / (m+n)

Theorems

Coordinate Geometry Theorem for Line Segment Division

Suitable Grade Level

Grades 9-11