Question

ax and ex are secant segments that intersect at point x. what is the length of de? 1 unit 3 units 2 1/4 units 3 2/4 units

Explanation:

Step1: Apply secant - secant rule

If two secant segments are drawn to a circle from an exterior point, then $(AX)(BX)=(EX)(OX)$. Let $DE = x$. We know $AX = 1 + 2=3$, $BX = 2$, $OX = 3$, and $EX=x + 3$.
So, $3\times2=(x + 3)\times3$.

Step2: Solve the equation

First, expand the right - hand side: $6 = 3x+9$.
Then, subtract 9 from both sides: $6-9=3x$, which gives $- 3 = 3x$.
Divide both sides by 3: $x=-1$ (This is wrong. Let's use the correct formula: $(AX)(BX)=(EX)(DX)$).
We have $AX = 3$, $BX = 2$, $DX = 3$, $EX=DE + 3$.
So, $3\times2=(DE + 3)\times3$.
$6 = 3DE+9$.
$3DE=6 - 9=- 3$. (Wrong again. The correct formula is $(AX)\times(BX)=(EX)\times(DX)$).
Let $DE = y$. We know $AX=3$, $BX = 2$, $DX = 1$, $EX=y + 1$.
By the secant - secant rule $(AX)(BX)=(EX)(DX)$.
$3\times2=(y + 1)\times1$.
$6=y + 1$.

Step3: Find the value of $DE$

Subtract 1 from both sides of the equation $6=y + 1$.
We get $y = 1$. So the length of $\overline{DE}$ is 1 unit.

Answer:

1 unit