Math Problem Statement

Point G lies on EF such that EG:FG is 1:3. Graph G.

Solution

To solve the problem of finding point GG on the line segment EFEF such that EG:FG=1:3EG:FG = 1:3, we can follow these steps:

Step 1: Determine the coordinates of points EE and FF

  • Point EE has coordinates (1,8)(1, -8).
  • Point FF has coordinates (6,4)(6, 4).

Step 2: Apply the section formula

Since GG divides EFEF in the ratio 1:31:3, we use the section formula to find the coordinates of GG. The section formula for a point dividing a line segment internally in the ratio m:nm:n is given by:

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

Here:

  • m=1m = 1
  • n=3n = 3
  • (x1,y1)=(1,8)(x_1, y_1) = (1, -8) (coordinates of EE)
  • (x2,y2)=(6,4)(x_2, y_2) = (6, 4) (coordinates of FF)

Step 3: Calculate the coordinates of GG

xG=(16)+(31)1+3=6+34=94=2.25x_G = \frac{(1 \cdot 6) + (3 \cdot 1)}{1 + 3} = \frac{6 + 3}{4} = \frac{9}{4} = 2.25

yG=(14)+(38)1+3=4244=204=5y_G = \frac{(1 \cdot 4) + (3 \cdot -8)}{1 + 3} = \frac{4 - 24}{4} = \frac{-20}{4} = -5

So, the coordinates of point GG are (2.25,5)\left(2.25, -5\right).

Step 4: Plot the point GG

On the graph, locate the point GG at (2.25,5)(2.25, -5). This point should lie on the line segment EFEF.

Would you like further details, or have any questions on this solution?

Here are some related questions you might explore:

  1. How does the section formula apply when GG divides EFEF externally?
  2. What would be the coordinates of GG if the ratio EG:FGEG:FG was 2:32:3?
  3. How can the midpoint formula be derived from the section formula?
  4. If point GG divided EFEF in a different ratio, how would you calculate it?
  5. How does the concept of weighted averages relate to the section formula?

Tip: Understanding the section formula can be particularly useful when solving problems involving line segments and ratios, especially in coordinate geometry.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Ratios
Section Formula

Formulas

Section formula for internal division G(x, y) = ((mx2 + nx1) / (m+n), (my2 + ny1) / (m+n))

Theorems

Section Theorem

Suitable Grade Level

Grades 9-10