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 $$ \frac {12 + 9 \sqrt {15} }{5} $$

I.S.I. 2017 geometry problem

Discussion:

This is a simple application of the** power of a point. ** We have to find the power of point A.

Suppose B = (0,0), C = (x,0) and D = (0,y).

Since AD = 5, therefore, coordinate of A = (0, 5+y).

Using the property of the power of point A, we have: ( AD \times AB = AE \times AC ) . Since AE is 4, this implies

( 5 \times (5+y) = 4 \times \sqrt { x^2 + (5+y)^2 } \ \Rightarrow 25 (5+y)^2 = 16 (x^2 + (5+y)^2 ) \ \Rightarrow 9(5+y)^2 = 16x^2 )

Since x and y are positive, we can take square roots on both sides. Hence we will have:

( 3(5+y) = 4x \ \Rightarrow y = \frac {4x}{3} – 5 )

Now note that ( \angle DBC = 90^o ) implying DC is a diameter. Hence DC is 12 (as radius is given to be 6).

Hence ( x^2 + y^2 = 12^2 ).

Replacing y by ( \frac {4x}{3} – 5 ) as found earlier, we have

( \displaystyle {x^2 + \left (\frac {4x}{3} – 5 \right )^2 = 12^2 \ \Rightarrow 9x^2 + 16x^2 – 120x + 225 = 144 \times 9 \ \Rightarrow 25x^2 – 120x – 1071 = 0 \ \Rightarrow x = \frac{120 \pm \sqrt { 120^2 + 107100}}{50} \ \Rightarrow x = \frac {120 \pm \sqrt { 14400 + 107100}}{50} \ \Rightarrow x = \frac {120 \pm 10 \sqrt {1215}}{50} \ \Rightarrow x = \frac {12 \pm 9 \sqrt {15}}{5} } )

Since x is positive we have (\displaystyle { x = \frac {12 + 9 \sqrt {15}}{5} } )

(proved)

Back to the question paper

*Related*

There is a more elegant solution available. Using secant theorem sinc=4/5

Hence cosc=3/5.

Also dec is right angled at E so is DEA. In triangle DEA using pythagoras DE=3

In triangle DEC, using pythagoras EC= 3root15.

Hence BC=AC cosc= (4+3root15)*3/5

Proved

sir i have a solution just using classic euclidean geometry.how to send ?

DEC is right angled at E so is DEA. In triangle DEA using Pythagoras DE=3.

NOW ADE & ABC ARE SIMILAR SO , ARE AD/DE=AC/BC. AND AC=5+ROOT(12^2+5^2)=>4+3ROOT15

USING THE RATIO WE GET 4+3ROOT15/BC=5/3 BC=(12+9ROOT15)/5