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

When two chords \(AC\) and \(BD\) intersect at point \(E\) in a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. So, \(AE\times EC=BE\times ED\). Substitute \(AE = x\), \(BE=x + 2\), \(ED=x + 5\), and \(EC=x + 12\) into the formula: \(x(x + 12)=(x + 2)(x + 5)\).

Step2: Expand both sides of the equation

For the left - hand side: \(x(x + 12)=x^{2}+12x\). For 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 (since \(x^{2}-x^{2}=0\)), we get \(12x=7x + 10\). Then subtract \(7x\) from both sides: \(12x-7x=7x + 10-7x\), which simplifies to \(5x=10\). Divide both sides by \(5\): \(x=\frac{10}{5}=2\).

Step4: Calculate the length of \(BD\)

Since \(BD=BE + ED\), and \(BE=x + 2\), \(ED=x + 5\). Substitute \(x = 2\) into the expressions. \(BE=2 + 2=4\) and \(ED=2+5 = 7\). Then \(BD=4 + 7=11\).

Answer:

\(11\)