Question

determining lengths of intersecting chord segments
be is 2 units longer than ae, de is 5 units longer than ae, and ce is 12 units longer than ae.
what is bd?
______ units

Explanation:

Step1: Apply intersecting chords theorem

For intersecting chords $AC$ and $BD$ at point $E$, the theorem states $AE \times CE = BE \times DE$. Substitute the given expressions:
$x(x + 12) = (x + 2)(x + 5)$

Step2: Expand both sides of the equation

Left side: $x^2 + 12x$
Right side: $x^2 + 5x + 2x + 10 = x^2 + 7x + 10$

Step3: Simplify to solve for $x$

Subtract $x^2 + 7x$ from both sides:
$x^2 + 12x - x^2 - 7x = 10$
$5x = 10$
$x = \frac{10}{5} = 2$

Step4: Calculate length of $BD$

$BD = BE + DE = (x + 2) + (x + 5)$. Substitute $x=2$:
$BD = (2 + 2) + (2 + 5)$

Answer:

11 units