Understanding the Section Formula for Ratio Calculation

This problem asks us to find the ratio in which a given point C divides the line segment connecting two other points A and B. To solve this, we will use the section formula, which helps determine the coordinates of a point that divides a line segment in a specific ratio.

Let the coordinates of point A be \((x_1, y_1) = (-2, -2)\).
Let the coordinates of point B be \((x_2, y_2) = (2, -4)\).
Let the coordinates of point C be \((x, y) = \left( - \frac{2}{7}, - \frac{20}{7} \right)\).

Suppose point C divides the line segment AB internally in the ratio \(m_1 : m_2\). The section formula for internal division is given by:

\(C(x, y) = \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right)\)

Applying the Section Formula to Find the Ratio

We can use either the x-coordinate or the y-coordinate of point C to find the ratio \(m_1 : m_2\). Let's use the x-coordinate first.

The x-coordinate of C is \(x = - \frac{2}{7}\). According to the section formula, this is equal to \(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}\).

So, we have:

\(- \frac{2}{7} = \frac{m_1 (2) + m_2 (-2)}{m_1 + m_2}\)
\(- \frac{2}{7} = \frac{2m_1 - 2m_2}{m_1 + m_2}\)

Now, we can cross-multiply:

\(-2 (m_1 + m_2) = 7 (2m_1 - 2m_2)\)
\(-2m_1 - 2m_2 = 14m_1 - 14m_2\)

Collect terms involving \(m_1\) on one side and terms involving \(m_2\) on the other side:

\(-2m_1 - 14m_1 = -14m_2 + 2m_2\)
\(-16m_1 = -12m_2\)

Now, we can find the ratio \(\frac{m_1}{m_2}\):

\(\frac{m_1}{m_2} = \frac{-12}{-16}\)
\(\frac{m_1}{m_2} = \frac{12}{16}\)
\(\frac{m_1}{m_2} = \frac{3}{4}\)

So, the ratio \(m_1 : m_2\) is 3 : 4.

Verifying the Ratio Using the Y-coordinate

We can also use the y-coordinate of C to verify this ratio. The y-coordinate of C is \(y = - \frac{20}{7}\). According to the section formula, this is equal to \(\frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\).

\(- \frac{20}{7} = \frac{m_1 (-4) + m_2 (-2)}{m_1 + m_2}\)
\(- \frac{20}{7} = \frac{-4m_1 - 2m_2}{m_1 + m_2}\)

Cross-multiply:

\(-20 (m_1 + m_2) = 7 (-4m_1 - 2m_2)\)
\(-20m_1 - 20m_2 = -28m_1 - 14m_2\)

Collect terms involving \(m_1\) on one side and terms involving \(m_2\) on the other:

\(-20m_1 + 28m_1 = -14m_2 + 20m_2\)
\(8m_1 = 6m_2\)

Find the ratio \(\frac{m_1}{m_2}\):

\(\frac{m_1}{m_2} = \frac{6}{8}\)
\(\frac{m_1}{m_2} = \frac{3}{4}\)

Both the x-coordinate and the y-coordinate give the same ratio, 3 : 4. Therefore, the point C divides the line segment joining A and B in the ratio 3 : 4.

Additional Information on Section Formula

The section formula is a fundamental concept in coordinate geometry. It can be used for both internal and external division of a line segment.

• Internal Division: When the point C lies between points A and B, it is an internal division. The ratio \(m_1 : m_2\) is positive.
• External Division: When the point C lies outside the line segment AB (on the line extending AB), it is an external division. The formula is similar, but with a minus sign: \(P(x, y) = \left( \frac{m_1 x_2 - m_2 x_1}{m_1 - m_2}, \frac{m_1 y_2 - m_2 y_1}{m_1 - m_2} \right)\). In such cases, the ratio \(m_1 : m_2\) is often considered negative or one of the terms in the numerator/denominator is negative depending on the convention used. Since our calculated ratio is positive, the division is internal.

The ratio is always stated as \(m_1 : m_2\), where \(m_1\) corresponds to the part of the segment connected to the second point B, and \(m_2\) corresponds to the part connected to the first point A. Specifically, the point C divides AB such that AC : CB = \(m_1 : m_2\). The point C is \(\frac{m_1}{m_1+m_2}\) fraction of the way from A to B.