Question

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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

The Intersecting Chords Theorem states that if two chords \(AB\) and \(CD\) intersect at point \(E\), then \(AE \times CE = BE \times DE\). Let \(AE = x\). Then \(BE = x + 2\), \(DE = x + 5\), and \(CE = x + 12\). Substituting into the theorem:
\[
x(x + 12)=(x + 2)(x + 5)
\]

Step2: Expand Both Sides

Expand the left - hand side: \(x(x + 12)=x^{2}+12x\)
Expand the right - hand side: \((x + 2)(x + 5)=x^{2}+5x+2x + 10=x^{2}+7x + 10\)
So the equation becomes:
\[
x^{2}+12x=x^{2}+7x + 10
\]

Step3: Solve for \(x\)

Subtract \(x^{2}\) from both sides: \(12x=7x + 10\)
Subtract \(7x\) from both sides: \(12x-7x=10\), i.e., \(5x = 10\)
Divide both sides by 5: \(x=\frac{10}{5}=2\)

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

We know that \(BE=x + 2\), substituting \(x = 2\), we get \(BE=2 + 2=4\)
We know that \(DE=x + 5\), substituting \(x = 2\), we get \(DE=2+5 = 7\)

Step5: Find the length of \(BD\)

Since \(BD=BE + DE\) (because \(E\) is a point on \(BD\)), then \(BD=4 + 7=11\)

Answer:

11