BdMO National Secondary 2020 P4

[Mursalin]

\(ABCD\) একটি বর্গক্ষেত্র। \(P\) এবং \(Q\) যথাক্রমে \(BC\) এবং \(CD\) রেখাংশের উপর দুইটি বিন্দু যাতে করে \(\angle APQ = 90^\circ\) হয়। দেওয়া আছে যে, \(AP = 4\) এবং \(PQ= 1\)। যদি \(AB\)-এর দৈর্ঘ্যকে লঘিষ্ঠ আকারে \(\frac{m}{n}\) হিসেবে লেখা যায়, তবে \(m+10n\)-এর মান বের কর।


\(ABCD\) is a square. \(P\) and \(Q\) are two points in segment \(BC\) and \(CD\) respectively such that \(\angle APQ = 90^\circ\). It is given that \(AP = 4\) and \(PQ = 1\). If we express the length of segment \(AB\) as \(\frac{m}{n}\) in lowest terms, compute \(m+10n\).

[Mehrab4226]

This problem can be solved by making three equations with three variables. But I am still looking for an easier solution.

[Ridwan056]

I think we can use similar triangles. Triangle APB and triangle PCQ are similar. Hence, AB= 4CP, which implies BP=3CP; and BP=4CQ. so, AP^2= 16= AB^2 +BP^2. Similarly, applying like this on PCQ gives BP=12/5. Therefore, AB=16/5. The final answer is m+10n= 16+50=66.