Math Problem Statement

given point C is (-1, 7) and D is (5,1), find 1/3 of the way from D to C

Solution

To find the point that is 13\frac{1}{3} of the way from point D to point C, we can use the section formula. The coordinates of point C are (1,7)(-1, 7) and point D are (5,1)(5, 1).

Using the formula for dividing a line segment in the ratio m:nm:n:

(mx2+nx1m+n,my2+ny1m+n)\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)

Here, m=1m = 1 (the part we want) and n=2n = 2 (the remaining part), giving us the ratio 1:21:2.

Substituting the coordinates of points C and D:

  • x1=5x_1 = 5, y1=1y_1 = 1
  • x2=1x_2 = -1, y2=7y_2 = 7

Now, substituting these values into the formula:

x=1(1)+251+2=1+103=93=3x = \frac{1 \cdot (-1) + 2 \cdot 5}{1+2} = \frac{-1 + 10}{3} = \frac{9}{3} = 3

y=17+211+2=7+23=93=3y = \frac{1 \cdot 7 + 2 \cdot 1}{1+2} = \frac{7 + 2}{3} = \frac{9}{3} = 3

Thus, the point that is 13\frac{1}{3} of the way from D to C is (3,3)(3, 3).

Summary:

The point 13\frac{1}{3} of the way from D to C is (3,3)(3, 3).

Would you like further details or have any questions? Here are some related questions to consider:

  1. What is the distance between points C and D?
  2. How do you calculate the midpoint between two points?
  3. Can you explain the section formula in more detail?
  4. What happens if you want to find a point halfway between C and D?
  5. How can you apply this concept to three-dimensional points?

Tip: Always double-check the ratios when using the section formula to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Section Formula

Formulas

Section formula: (mx2 + nx1) / (m+n), (my2 + ny1) / (m+n)

Theorems

Section theorem

Suitable Grade Level

Grades 9-12