If a vertex of a triangle is (1, 1) and the mid points of two sides through this vertex are (-1, 2)

Q: Wait, can't I just find the midpoint of the two given midpoints to get the centroid?

That is a very common trap! The midpoint of the two given midpoints is actually the midpoint of the segment connecting them, which is parallel to the third side of the triangle. It is not the centroid. The centroid is the center of mass of the entire triangle, which requires all three vertices.

Q: Is there a shortcut? Do I really need to find vertices $B$ and $C$?

While you could use the property that the centroid of the triangle formed by the midpoints is the same as the centroid of the original triangle, finding the vertices $B$ and $C$ is a fundamental skill that prevents errors. In JEE, clarity is speed. Stick to the method that guarantees accuracy.

Q: What if I made a sign error while calculating $2M - A$?

Sign errors are the silent killers of JEE scores. Always write out the coordinates clearly: $2(-1, 2) = (-2, 4)$, then subtract $(1, 1)$. Taking that extra second to write it down is the difference between a correct answer and a negative marking.

Question

If a vertex of a triangle is $(1, 1)$ and the mid points of two sides through this vertex are $(-1, 2)$ and $(3, 2)$ then the centroid of the triangle is

Options:

(A) $(-1, \frac{7}{3})$

(B) $(\frac{-1}{3}, \frac{7}{3})$

(C) $(1, \frac{7}{3})$

(D) $(\frac{1}{3}, \frac{7}{3})$

Sigma Percentile · Accepted Answer

### The Geometry of Balance

Imagine you are standing at the vertex $A(1, 1)$ of a triangle. You are given two midpoints, $M_1(-1, 2)$ and $M_2(3, 2)$, which lie on the sides connected to your vertex.

This is a classic coordinate geometry puzzle. It is not just about plugging numbers into formulas; it is about understanding the hidden symmetry of a triangle.

The midpoint formula, $M = \frac{A+B}{2}$, is your bridge to the unknown. It tells us that the midpoint is the average of the two endpoints. If we know the average and one endpoint, we can reverse the process to find the other endpoint.

### Reverse Engineering the Vertices

We know that $M_1$ is the midpoint of $AB$. Therefore, $M_1 = \frac{A+B}{2}$. By rearranging this, we get $B = 2M_1 - A$.

Let us calculate this carefully. We take $2 	imes (-1, 2)$, which gives us $(-2, 4)$. Now, we subtract the coordinates of $A(1, 1)$.

So, the $x$-coordinate of $B$ is $-2 - 1 = -3$, and the $y$-coordinate is $4 - 1 = 3$. Thus, vertex $B$ is at $(-3, 3)$.

We apply the same logic for vertex $C$ using $M_2(3, 2)$ where $C = 2M_2 - A$. This gives us $2 	imes (3, 2) = (6, 4)$.

Subtracting $A(1, 1)$ yields $6 - 1 = 5$ and $4 - 1 = 3$. So, vertex $C$ is at $(5, 3)$. We have successfully reconstructed the triangle!

### Finding the Centroid

Now that we have all three vertices—$A(1, 1)$, $B(-3, 3)$, and $C(5, 3)$—we can find the centroid $G$. The centroid is the geometric center of the triangle, the point where the three medians intersect.

It is simply the average of the $x$-coordinates and the average of the $y$-coordinates:

$$G = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} ight)$$

Substituting our values, we get:

$$G = \left( \frac{1 + (-3) + 5}{3}, \frac{1 + 3 + 3}{3} ight)$$

Simplifying the numerators, we find $1 - 3 + 5 = 3$ and $1 + 3 + 3 = 7$. Therefore, the centroid $G$ is $\left( \frac{3}{3}, \frac{7}{3} ight)$, which simplifies to:

**$G = \left( 1, \frac{7}{3} ight)$**

This is the elegant balance point of our triangle. Remember, in JEE, always double-check your arithmetic, especially with negative signs. You have mastered the logic; now trust your process.

Animated Solution for Mathematics - Straight Lines: If a vertex of a triangle is $(1, 1)$ and the mid points of two sides through this vertex are $(- 1, 2)$ and $(3, 2)$ then the centroid of the triangle is

Select Answer:

Visualized Solution

Visualizing the Given Points

The Midpoint Formula Logic

Setting up Vertex $B$

Calculating Vertex $B$

Setting up Vertex $C$

Calculating Vertex $C$

The Centroid Formula

Substituting Coordinates

Final Centroid Calculation

Summary and Key Takeaways

The Sigma Insight: Centroid, Incenter, Orthocenter, and Circumcenter

The Geometry of Balance

Reverse Engineering the Vertices

Finding the Centroid

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