Try this beautiful problem from PRMO, 2012 based on Triangle.
In \(\triangle ABC\) we have \(AC=BC=7\) and \(AB=2\).Suppose that \(D\) is a point on line \(AB\) such that \(B\) lies between \(A\) and \(D\) and \(CD=8\) .what is the length of the segment \(DB\)?
Geometry
Triangle
Pythagoras
Answer:\(3\)
PRMO-2012, Problem 7
Pre College Mathematics
Given that \(AC=BC=7\) & \(CD=8\).we have to find out \(BD\).Let \(BD=x\).we draw a perpendicular from \(C\) to \(AB\) at the point \(M\).Therefore clearly \(\triangle CMB\) & \(\triangle CMD\) are right angle.so if we can find out the value of \(CM\) from the \(\triangle CMB\) then we can find out the value \(BD\) from the \(\triangle CMD\)
Can you now finish the problem ..........
From the above diagram,In \(\triangle CMB\) we can say that \(CM=\sqrt{49-1}=4\sqrt 3\)
Given \(AB=2\) and \(M\) is the mid point of \(\triangle ABC\) (As AC=BC=7,Isosceles triangle),
Therefore \(BM=1\), So \(MD=x+1\)
From the \(\triangle CMD\), \((X+1)^2+(4\sqrt 3)^2=64\) \(\Rightarrow x=3,-5\)
we will take the positive value ,so \(BD=3\)
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.