Question

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 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