rdrdrd1201 wrote:
Dillesh4096 wrote:
Bunuel wrote:
Competition Mode Question
In a circle of radius 11 cm, CD is a diameter and AB is a chord of length 20.5 cm. If AB and CD intersect at a point E inside the circle and CE has length 7 cm, then the difference of the lengths of BE and AE, in cm, is
A. 0.5
B. 1.5
C. 2
D. 2.5
E. 3.5
Rule: For any 2 chords AB & CD of a circle intersecting at point E, \(AE*BE\) = \(CE*DE\)From the figure, \(AE\) = \(x\), \(BE\) = \(20.5 - x\), \(CE\) = \(7\) & \(DE\) = \(15\)
--> \(AE*BE\) = \(7*15\) = \(105\)
& \(AE\) + \(BE\) = \(20.5\)
--> \((AE - BE)^2 = (AE + BE)^2 - 4*AE*BE\)
--> \((AE - BE)^2 = (20.5)^2 - 4*105 = 420.25 - 420 = 0.25\)
--> \(AE - BE = \sqrt{0.25} = 0.5\)
Option A
hi dilesh can you explain how you get to:
--> (AE−BE)2=(AE+BE)2−4∗AE∗BE(AE−BE)2=(AE+BE)2−4∗AE∗BE
--> (AE−BE)2=(20.5)2−4∗105=420.25−420=0.25(AE−BE)2=(20.5)2−4∗105=420.25−420=0.25
--> AE−BE=0.25−−−−√=0.5
here?
i dont get where the equation is coming from
Think of AE as x and BE as y, where \(x + y = 20.5\); what we're looking for is \(|x-y|\). Now remember the properties that:
\((x - y)^2 = x^2 - 2xy + y^2\)
\((x + y)^2 = x^2 + 2xy + y^2\)
To get \((x - y)^2\) from \((x + y)^2\), we have to subtract \(4xy\) to get:
\((x + y)^2 - 4xy = x^2 + 2xy + y^2 - 4xy = x^2 - 2xy + y^2 = (x - y)^2\).
Therefore, we can solve as follows:
\((x - y)^2 = (x + y)^2 - 4xy\)
-> \((x - y)^2 = (20.5)^2 - 4(105)\) (because \(AE * BE = CE * DE = 7 * 15 = 105\) by the Intersecting Chords Theorem)
-> \((x - y)^2 = 0.25\)
-> \((x - y) = \sqrt{0.25} = 0.5\) or \(-0.5\)
-> \(|x - y| = 0.5\)
Answer choice A. A great solution by Bunuel, hope this explanation helps you, it took me a while to also understand this question!