How to Prove a Line Divides Another Line Segment Internally in a Given Ratio

In this article, we will explore the concept of proving that a line segment A0 -1 B15 2 divides another line segment C-1 2 D4 -5 internally in a specific ratio, in this case, 2:3. This process involves several steps: finding the equations of the lines, identifying the point of intersection, and then checking the ratio. We will walk through each step in detail.

Step 1: Find the Equations of the Lines

Firstly, we need to find the equations of the lines AB and CD.

Line AB

The coordinates of point A are (0, -1) and of point B are (15, 2).

The slope mAB is calculated as:

mAB (2 - (-1)) / (15 - 0) 3 / 15 1 / 5

The equation of line AB can be derived using the point-slope form:

y - (-1) (1/5)(x - 0)

Simplifying this, we get:

y (1/5)x - 1

Line CD

The coordinates of point C are (-1, 2) and of point D are (4, -5).

The slope mCD is given by:

mCD (-5 - 2) / (4 - (-1)) -7 / 5

The equation of line CD is:

y - 2 (-7/5)(x 1)

Rearranging, we get:

y - (7/5)x - 7/5 2 - (7/5)x 3/5

Step 2: Find the Point of Intersection

To find the intersection point, we set the equations of lines AB and CD equal to each other:

(1/5)x - 1 - (7/5)x 3/5

Multiplying through by 5 to eliminate fractions:

x - 5 -7x 3

8x 8

x 1

Substituting x 1 back into AB to find y:

y (1/5)(1) - 1 -4/5

The intersection point P is therefore (1, -4/5).

Step 3: Check the Ratio

We use the section formula to find the point dividing a line segment in a given ratio. For points C(-1, 2) and D(4, -5) with a ratio of 2:3:

The coordinates of the point P are:

x (2*4 3*(-1)) / (2 3) (8 - 3) / 5 5/5 1

y (2*(-5) 3*2) / (2 3) (-10 6) / 5 -4/5

The point P at (1, -4/5) confirms that the line AB divides the line segment CD internally in the ratio 2:3.

Conclusion

We have demonstrated through the steps that the line AB intersects the line segment CD internally at the point (1, -4/5) and divides it in the ratio 2:3. This method can be applied to similar problems in coordinate geometry to verify line division by ratio.