Problem 116

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Problem 116: (2024 AMC 12b, 20)

Suppose $A$, $B$, and $C$ are points in the plane with $AB=40$ and $AC=42$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p,q)$, and the maximum value $r$ of $f(x)$ occurs at $x=s$. What is $p+q+r+s$? $\textbf{(A) }909\qquad$ $ \textbf{(B) }910\qquad$ $ \textbf{(C) }911\qquad$ $ \textbf{(D) }912\qquad$ $ \textbf{(E) }913\qquad$

Solution

Exercise 116: Solution

In $\triangle ABC$, $AB=10$ and $AC=24$. Let $x$ be the length of the segment from $A$ to the midpoint of $\overline{BC}$, and define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p,q)$, and the maximum value $r$ of $f(x)$ occurs at $x=s$. What is $p+q+r+s$?

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