Coordinate Geometry
Angle at vertex subtended by segment of hypotenuse
MJAT_TS5_P2
Grade 12
Question:
Let $A(0,0)$, $B(b,0)$, $C(0,c)$ be vertices of a triangle with $BC=a$. $BC$ is divided into $n$ equal parts ($n$ odd). Let $\theta$ be the angle subtended at $A$ by the segment containing the midpoint of $BC$. If $h$ is the length of the altitude to the hypotenuse and $\tan\theta=\dfrac{pnh}{(qn^2-r)a}$ where $p,q,r\in\mathbb{N}$ and $\gcd(p,q,r)=1$, then which is/are correct?
A) $p>q$
B) $p<q$
C) $p+q=5$
D) $r=2$
Step-by-Step Solution
Key Concept: The midpoint of $BC$ is $M=(b/2,c/2)$. The segment around $M$ has endpoints at distances $(n\pm 1)/(2n)$ along $BC$ from $B$. The angle at $A$ subtended by this segment can be computed using the tangent difference formula.
$p=4$, $q=1$, $r=1$. A ✓ ($p>q$), C ✓ ($p+q=5$). Answer: A, C.
Correct Answer: AC