Math Problem Statement

The point B lies on the segment AC. Find the coordinates of B so that the ratio of AB to BC is 4 to 3. Given points A(-4, -6) and C(17, 22).

Solution

To solve the problem in the image, we need to find the coordinates of point B(x,y)B(x, y) such that the ratio of ABAB to BCBC is 4:34:3.

Step 1: Understand the Coordinates

Given:

  • A(4,6)A(-4, -6)
  • C(17,22)C(17, 22)
  • Ratio AB:BC=4:3AB : BC = 4 : 3

Step 2: Apply the Section Formula

The section formula in coordinate geometry is used to find the coordinates of a point dividing a line segment in a given ratio. If a point B(x,y)B(x, y) divides the line segment joining A(x1,y1)A(x_1, y_1) and C(x2,y2)C(x_2, y_2) in the ratio m:nm:n, then the coordinates of BB are given by:

x=mx2+nx1m+nx = \frac{mx_2 + nx_1}{m+n} y=my2+ny1m+ny = \frac{my_2 + ny_1}{m+n}

Here, m=4m = 4 and n=3n = 3, A(x1,y1)=(4,6)A(x_1, y_1) = (-4, -6), C(x2,y2)=(17,22)C(x_2, y_2) = (17, 22).

Step 3: Substitute the Values

x=417+3(4)4+3=68127=567=8x = \frac{4 \cdot 17 + 3 \cdot (-4)}{4 + 3} = \frac{68 - 12}{7} = \frac{56}{7} = 8 y=422+3(6)4+3=88187=707=10y = \frac{4 \cdot 22 + 3 \cdot (-6)}{4 + 3} = \frac{88 - 18}{7} = \frac{70}{7} = 10

Final Answer

The coordinates of point BB are (8,10)(8, 10).

Would you like more details or have any questions?

Related Questions:

  1. How is the section formula derived in coordinate geometry?
  2. What are other methods to find a point dividing a line segment?
  3. Can you explain the midpoint formula as a special case of the section formula?
  4. How would the coordinates change if the ratio was reversed to 3:4?
  5. How can you verify the answer using the distance formula?

Tip:

The section formula is a powerful tool not only for dividing line segments in a specific ratio but also for solving many geometric problems involving coordinate points.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Ratios

Formulas

Section Formula: x = (mx2 + nx1) / (m+n), y = (my2 + ny1) / (m+n)

Theorems

Section Formula

Suitable Grade Level

Grades 9-12