Math Problem Statement

Find the coordinates of the point 3/10 of the way from A(-3, -5) to B(9, 4).

Solution

We will use the section formula to find the point that is 310\frac{3}{10} of the way from AA to BB.

Given:

  • A(3,5)A(-3, -5)
  • B(9,4)B(9,4)
  • Ratio: 310\frac{3}{10} of the way from AA to BB
    • This means the division ratio is 3:73:7 (since 3+7=103+7=10).

The section formula states that a point dividing the line segment joining A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2) in the ratio m:nm:n is given by:

P(mx2+nx1m+n,my2+ny1m+n)P\left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)

Substituting values: Px=(3×9)+(7×3)3+7=272110=610=0.6P_x = \frac{(3 \times 9) + (7 \times -3)}{3+7} = \frac{27 - 21}{10} = \frac{6}{10} = 0.6

Py=(3×4)+(7×5)3+7=123510=2310=2.3P_y = \frac{(3 \times 4) + (7 \times -5)}{3+7} = \frac{12 - 35}{10} = \frac{-23}{10} = -2.3

Final Answer:

The coordinates of the required point are (0.6,2.3)(0.6, -2.3).

Would you like a detailed explanation of any step?

Related Questions:

  1. How do you use the section formula for different ratios?
  2. What is the midpoint formula, and how does it relate to the section formula?
  3. How do you find a point dividing a line segment externally?
  4. Can the section formula be applied in 3D geometry?
  5. How can we verify the correctness of the calculated point graphically?

Tip:

Always check the ratio before applying the section formula to ensure the correct placement of the point.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Ratios
Linear Interpolation

Formulas

Section formula: P((m*x2 + n*x1)/(m+n), (m*y2 + n*y1)/(m+n))

Theorems

Section formula theorem

Suitable Grade Level

Grades 8-10