In $\triangle ABC$, coordinates of $A$ are $(1, 2)$, and the equations of the medians through $B$ and $C$ are $x + y = 5$ and $x = 4$, respectively. Then the midpoint of $BC$ is:

Question

cdquestions Admin · Accepted Answer

The median through $A(1, 2)$ passes through the midpoint of side $BC$. Let the midpoint of $BC$ be $(x, y)$. The median through $B$ is given by $x + y = 5$. The median through $C$ is given by $x = 4$. Step 1: Solve for $x$. From the equation of the median through $C$, we know: $$ x = 4. $$ Step 2: Solve for $y$. Substitute $x = 4$ into the equation of the median through $B$: $$ 4 + y = 5 \implies y = 1. $$ Thus, the midpoint of $BC$ is: $$ (4, 1). $$ Step 3: Adjust for the centroid. The centroid divides the median in the ratio $2 : 1$. Since $A$ is at $(1, 2)$, the coordinates of the midpoint of $BC$ are: $$ \left(\frac{11}{2}, \frac{1}{2}\right). $$ Conclusion: The midpoint of $BC$ is: $$ \boxed{\left(\frac{11}{2}, \frac{1}{2}\right)}. $$

Answer

$\left(5, \frac{1}{2}\right)$

Answer

$\left(\frac{11}{2}, 1\right)$

Answer

$\left(11, \frac{1}{2}\right)$

Answer

$\left(\frac{11}{2}, \frac{1}{2}\right)$

In \(\triangle ABC\), coordinates of \(A\) are \((1, 2)\), and the equations of the medians through \(B\) and \(C\) are \(x + y = 5\) and \(x = 4\), respectively. Then the midpoint of \(BC\) is:

The Correct Option is D

Solution and Explanation

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