solving for x

[andy849]

In a diagram of circle, chords AB and CD
intersect at E. If AE = 3, EB = 4, CE = x, and ED = x - 4, what is the
value of x?

how do I start this problem?

 

[soroban]

Hello, andy849!

$\displaystyle \text{In a diagram of a circle, chords }AB \text{ and }CD\text{ intersect at }E.$

$\displaystyle \text{If }\,AE \,=\, 3,\; EB \,=\, 4,\; CE \,=\, x,\;ED \,=\, x - 4,\,\text{ what is the value of }x\,?$

Theorem: If two chords intersect inside a circle, the products of their segments are equal.


$\displaystyle \text{So we have: }\;x(x-4) \:=\:3\cdot4 \quad\Rightarrow\quad x^2 - 4x - 12 \:=\:0$

$\displaystyle \text{Hence: }\;(x-6)(x+2) \:=\:0 \quad\Rightarrow\quad x \:=\:6,\;\rlap{\;///}-2$


$\displaystyle \text{Therefore: }\:x \,=\,6$