Animated Solution for Mathematics - Straight Lines: The equations of the sides $AB,BC$ and $CA$ of a triangle $ABC$ are $2x+y=0,x+py=39$ and $x-y=3$ respectively and $P(2,3)$ is its circumcentre. Then which of the following is NOT true :

We are tasked with reconstructing a triangle $ABC$ given the equations of its sides and the coordinates of its circumcenter $P(2,3)$. This problem requires us to treat the circumcenter as the geometric anchor from which all vertices must be equidistant.

Finding the Anchor Point

Vertex $A$ is the intersection of lines $AB$ ($2x+y=0$) and $CA$ ($x-y=3$). We solve these linear equations simultaneously:

Adding the two equations yields $3x=3$, which implies $x=1$. Substituting $x=1$ into $x-y=3$, we find $y=-2$.

Thus, our anchor, vertex $A$, is located at $(1,-2)$.

The Radius of Influence

The distance from the circumcenter $P(2,3)$ to any vertex is defined as the circumradius $R$. Using vertex $A(1,-2)$, we calculate $R^2$ as follows:

This value, $26$, serves as the constant squared distance that every vertex must maintain from $P$.

The Hunt for Vertices B and C

Vertex $B$ lies on the line $AB$ ($2x+y=0$), so we express $B$ as $(x,-2x)$. Given $P{B}^2=R^2=26$, we set up the following equation:

Expanding this expression, we obtain:

Factoring the quadratic, we get $(x-1)(5x+13)=0$. The roots are $x=1$ (which corresponds to vertex $A$) and $x=-\frac{13}{5}$. Therefore, vertex $B$ is at $(-\frac{13}{5},\frac{26}{5})$.

Next, we find vertex $C$, which lies on $x-y=3$, or $y=x-3$. Setting $P{C}^2=26$:

This simplifies to $2x^2-16x+14=0$, or $x^2-8x+7=0$. The roots are $1$ and $7$. Since $x=1$ is vertex $A$, vertex $C$ must be at $(7,4)$.

The Final Verification

With $B(-\frac{13}{5},\frac{26}{5})$ and $C(7,4)$ identified, we calculate the slope $m$ of line $BC$:

The equation of line $BC$ is $y-4=-\frac{1}{8}(x-7)$, which rearranges to $x+8y=39$. Comparing this to the form $x+py=39$, we find $p=8$.

Finally, we calculate the area of $\Delta ABC$ using the coordinate determinant formula:

Substituting our coordinates, we find the area to be $32.4$. We have successfully navigated the geometry and identified the properties of the triangle.