In-ABC-it-is-given-that-AC-CB-CD-AB-and-CD-12-AC-BC-5-Please-solve-for-the-value-of-BC-using-a-purely-geometric-method-

Question Number 215769 by momoga last updated on 17/Jan/25

In △ABC, it is given that AC⊥CB, CD⊥AB, and CD = 12, AC = BC + 5. Please solve for the value of BC using a purely geometric method.

Commented by momoga last updated on 17/Jan/25

Answered by A5T last updated on 17/Jan/25

AD = \sqrt{{(BC + 5)}^{2} - 12^{2}}

BD = \sqrt{{BC}^{2} - 12^{2}}

AD + BD = \sqrt{{(BC + 5)}^{2} + {BC}^{2}}

\Rightarrow {(AD + BD)}^{2} = {AD}^{2} + {BD}^{2} + 2 AD \times BD

\Rightarrow 144 = \sqrt{({BC}^{2} - 12^{2}) ({BC}^{2} + 10 BC - 119)}

\Rightarrow 144^{2} = (BC - 12) (BC + 12) (BC + 17) (BC - 7)

\Rightarrow BC = 15

Commented by momoga last updated on 18/Jan/25

Oh, so that’s how it is

Commented by momoga last updated on 17/Jan/25

I believe this is an algebraic approach, involving solving a quartic equation. Is there a geometric method, perhaps?

Commented by AntonCWX last updated on 18/Jan/25

N o .

W h e n w e a r e s o l v i n g t h i s k i n d o f p r o b l e m,

W e a p p l y c o n c e p t f r o m g e o m e t r y a n d t r a n s f o r m i t i n t o e q u a t i o n s .

Commented by AntonCWX last updated on 18/Jan/25

T h e o n l y p u r e g e o m e t r i c w a y i s t o d r a w t h e d i a g r a m i n s c a l e a n d m e a s u r e t h e l e n g t h .