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

We have a circle with radius 11 cm, which means:

CD is a diameter = 2 × radius = 2 × 11 = 22 cm

AB is a chord = 20.5 cm

These two lines intersect at point E inside the circle

CE = 7 cm

Since CD is the diameter (22 cm) and CE = 7 cm, then:

DE = CD - CE = 22 - 7 = 15 cm

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When two chords intersect inside a circle, there's a beautiful relationship:

AE × BE = CE × DE

This works because when chords intersect inside a circle, the triangles formed are similar due to equal angles (angles subtended by the same arc are equal). This similarity gives us the product relationship.

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Let's say AE = x

Since AB = 20.5 cm total, then:

BE = 20.5 - x

Using our intersecting chords theorem:

AE × BE = CE × DE

x × (20.5 - x) = 7 × 15

x(20.5 - x) = 105

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20.5x - x² = 105

x² - 20.5x + 105 = 0

Using the quadratic formula: $x = \frac{20.5 ± \sqrt{20.5² - 4×105}}{2}$

Let us calculate the discriminant:

20.5² - 4×105 = 420.25 - 420 = 0.25

So: $x = \frac{20.5 ± \sqrt{0.25}}{2} = \frac{20.5 ± 0.5}{2}$

This gives us two solutions:

x = 21/2 = 10.5 or x = 20/2 = 10

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If AE = 10.5, then BE = 20.5 - 10.5 = 10

If AE = 10, then BE = 20.5 - 10 = 10.5

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In both cases, the difference is:

|BE - AE| = |10 - 10.5| = 0.5 cm

Therefore, the difference of the lengths of BE and AE is 0.5 cm.

It doesn't matter which segment is AE and which is BE - the absolute difference remains the same!