In the given figure, point D divides the side BC of \(\triangle ABC\) in the ratio \(1 : 2\). Find length AD. (Given coordinates: \(A(1, 5), B(-2, 1), C(4, 2)\))
Show Hint
Always double-check the order of \(m\) and \(n\) in the section formula relative to points \(B\) and \(C\). Since the ratio is \(1:2\) from \(B\) to \(C\), \(m=1\) multiplies the coordinates of \(C\).
Step 1: Understanding the Concept:
To find the length of the segment \(AD\), we first need to find the coordinates of point \(D\) using the section formula.
Once the coordinates of \(D\) are known, we use the distance formula between points \(A\) and \(D\). Step 2: Key Formula or Approach:
Section Formula for a point dividing a line in ratio \(m:n\):
\[ D(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Distance Formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Step 3: Detailed Explanation:
Given coordinates: \(B(-2, 1)\), \(C(4, 2)\), and ratio \(m:n = 1:2\).
Let \(D\) have coordinates \((x, y)\).
\[ x = \frac{1(4) + 2(-2)}{1+2} = \frac{4 - 4}{3} = 0 \]
\[ y = \frac{1(2) + 2(1)}{1+2} = \frac{2 + 2}{3} = \frac{4}{3} \]
So, coordinates of \(D\) are \((0, \frac{4}{3})\).
Now, we find the length \(AD\) where \(A\) is \((1, 5)\):
\[ AD = \sqrt{(0 - 1)^2 + \left( \frac{4}{3} - 5 \right)^2} \]
\[ AD = \sqrt{(-1)^2 + \left( \frac{4 - 15}{3} \right)^2} \]
\[ AD = \sqrt{1 + \left( \frac{-11}{3} \right)^2} \]
\[ AD = \sqrt{1 + \frac{121}{9}} \]
\[ AD = \sqrt{\frac{9 + 121}{9}} = \sqrt{\frac{130}{9}} \]
\[ AD = \frac{\sqrt{130}}{3} \text{ units} \] Step 4: Final Answer:
The length of \(AD\) is \(\frac{\sqrt{130}}{3}\) units.
