Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
In a $\triangle ABC$, the equations of right bisectors of sides $AB$ and $AC$ are $3x + 4y = 20$ and $8x + 6y = 65$ respectively. If the vertex $A$ be $(10, 10)$, then the value of $\frac{1}{7}(ar\triangle ABC)$ is ________.
Step-by-Step Solution
Key Concept: The circumcenter is the intersection of perpendicular bisectors of two sides of a triangle.
The perpendicular bisectors of $AB$ and $AC$ are $3x + 4y - 20 = 0$ and $8x + 6y - 65 = 0$ respectively. Finding their intersection gives the circumcenter. Point $B$ lies on line $3x + 4y - 20 = 0$ such that its perpendicular bisector passes through it, yielding $B = (-2, -6)$. Similarly, $C = (-2, 1)$ is found. The area of triangle $ABC$ is calculated as $ rac{1}{2}(10 + 2)(1 + 6) = 42$, so the required value is $6$.
Correct Answer: 6