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…

Explanation:

Step1: Apply the intersecting - chords theorem

If two chords (AC) and (BD) intersect at a point (E) inside a circle, then (AE\times CE=BE\times DE). Given (AE = x), (BE=x + 2), (CE=x + 12), and (DE=x + 5). So, (x(x + 12)=(x + 2)(x + 5)).

Step2: Expand both sides of the equation

Expand the left - hand side: (x(x + 12)=x^{2}+12x). Expand the right - hand side: ((x + 2)(x + 5)=x^{2}+5x+2x + 10=x^{2}+7x + 10).

Step3: Solve the resulting linear equation

Set (x^{2}+12x=x^{2}+7x + 10). Subtract (x^{2}) from both sides of the equation: (12x=7x + 10). Then subtract (7x) from both sides: (12x-7x=10), so (5x = 10), and (x = 2).

Step4: Find the lengths of (BE) and (DE)

Since (x = 2), (BE=x + 2=2 + 2=4) and (DE=x + 5=2+5 = 7).

Step5: Calculate (BD)

(BD=BE + DE). So (BD=4 + 7=11).

Answer:

11