Select Page

# Understand the problem

Consider a circle of radius 6 as given in the diagram below. Let $$B,C,D$$ and $$E$$ be points on the circle such that $$BD$$ and $$CE$$, when extended, intersect at $$A$$. If $$AD$$ and $$AE$$ have length 5 and 4 respectively, and $$DBC$$ is a right angle, then show that the length of $$BC$$ is $$\frac{12+\sqrt{15}}{5}$$.

##### Source of the problem

I.S.I. (Indian Statistical Institute, B.Stat, B.Math) Entrance. Subjective Problem 2 from 2017

Plane Geometry

6 out of 10

##### Suggested Book

Do you really need a hint? Try it first!

Given, $$AD=5, AE=4$$ and $$\angle DBC=90^\circ$$.

As $$D,B,C$$ are points on the circle having radius 6.

Therefore $$DC$$ is the diameter of the circle

$$\Rightarrow DC=6×2=12$$.

Now DC is diameter of the circle $$\Rightarrow \angle DEC=90^\circ$$.

Therefore $$\angle DEA$$ is also right angle.

The length of $$DE=\sqrt{5^2-4^2}=3$$.

And the length of $$EC=\sqrt{12^2-3^2}=3\sqrt{15}$$.

Therefore $$AC=AE+EC=4+3\sqrt{15}$$.

From $$∆AED$$ and $$∆ABC$$ we have,

$$\frac{AD}{DE}=\frac{AC}{BC} \Rightarrow BC=\frac{AC\cdot DE}{AD}=\frac{(4+3\sqrt{15})\cdot 3}{5}$$.

Therefore the length of $$BC$$ is $$\frac{12+9\sqrt{15}}{5}$$. (Ans.)

# I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

# Similar Problems

## Solving a congruence

Understand the problemProve that the number of ordered triples in the set of residues of $latex p$ such that , where and is prime is . Brazilian Olympiad Revenge 2010 Number Theory Medium Elementary Number Theory by David Burton Start with hintsDo you really need...

## Inequality involving sides of a triangle

Understand the problemLet be the lengths of sides of a (possibly degenerate) triangle. Prove the inequalityLet be the lengths of sides of a triangle. Prove the inequalityCaucasus Mathematical Olympiad Inequalities Easy An Excursion in Mathematics Start with hintsDo...

## Vectors of prime length

Understand the problemGiven a prime number and let be distinct vectors of length with integer coordinates in an Cartesian coordinate system. Suppose that for any , there exists an integer such that all three coordinates of is divisible by . Prove that .Kürschák...

## Missing digits of 34!

Understand the problem34!=295232799cd96041408476186096435ab000000 Find $latex a,b,c,d$ (all single digits).BMO 2002 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!Get prepared to find the residue of 34! modulo...

## An inequality involving unknown polynomials

Understand the problemFind all the polynomials of a degree with real non-negative coefficients such that , . Albanian BMO TST 2009 Algebra Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!This problem is all about...

## Hidden triangular inequality (PRMO Problem 23, 2019)

Problem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...

## PRMO – 2019 – Questions, Discussions, Hints, Solutions

This is a work in progress. Please post your answers in the comment. We will update them here. Point out any error that you see here. Thank you. 1. 42. 133. 134. 725. 106. 297. 518. 499. 1410. 5511. 612. 1813. 1014. 5315. 4516. 4017. 3018. 2019. 1320. Bonus21. 1722....

## Bangladesh MO 2019 Problem 1 – Number Theory

A basic and beautiful application of Numebr Theory and Modular Arithmetic to the Bangladesh MO 2019 Problem 1.

## Functional equation dependent on a constant

Understand the problemFind all real numbers for which there exists a non-constant function satisfying the following two equations for all i) andii) Baltic Way 2016 Functional Equations Easy Functional Equations by BJ Venkatachala Start with hintsDo you really need...

## Pigeonhole principle exercise

Problem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...