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Geometry Level 3

$AB$ and $CD$ are perpendicular chords that intersect at $E$ .

If $CE = 3, AC = 5,$ and $ED = 12,$ what is the length of $BD$ ?

15 16 18 20

2 solutions

Jesse Nieminen
Oct 23, 2016

Relevant wiki: Power of a Point

Since $AB$ and $CD$ are perpendicular, $\bigtriangleup ACE$ and $\bigtriangleup BDE$ are right triangles.

Applying the Pythagorean theorem on $\bigtriangleup ACE$ gives $AE = \sqrt{AC^2 - CE^2} = 4$ .

Now, the two secants case of the power of a point theorem gives $EC \cdot ED = EA \cdot EB \Rightarrow EB = \dfrac{EC \cdot ED}{EA} = 9$

Hence, applying the Pythagorean theorem on $\bigtriangleup BDE$ gives $BD = \sqrt{ED^2 + EB^2} = \boxed{15}$ which is the answer.

A common mistake that I expect people to make, is that they remember $ACE$ and $BDE$ are similar triangles, and then think that the similarity ratio is $3: 12$ , hence give the corresponding length is $5 \times 4 = 20$ .

However, what we have to be careful about, is to pair up the corresponding lengths properly. In this case, CE and ED are not corresponding lengths. Instead, the corresponding pair is AE and ED, which leads to the similarity ratio of $4 : 12$ .

Calvin Lin Staff - 4 years, 7 months ago

Vishwash Kumar
Oct 31, 2016

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