Question

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as shown below, a big tree broke during a hurricane. after falling, the top of the tree (point a) is 12 m away from the base of the tree (point b). before breaking, the tree was 18 m tall. the distance from the break point to the ground (\\(\overline{cb}\\)) is \\(\square\\) m.

Explanation:

Step1: Set up variables

Let \(CB = x\) meters. Then \(AC=(18 - x)\) meters.

Step2: Apply Pythagorean theorem

In right - triangle \(ABC\), by the Pythagorean theorem \(AB^{2}+BC^{2}=AC^{2}\). Given \(AB = 12\) m. Substitute the values into the formula: \(12^{2}+x^{2}=(18 - x)^{2}\).

Step3: Expand the equation

Expand \((18 - x)^{2}\) using the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\), where \(a = 18\) and \(b=x\). So \(144+x^{2}=324-36x+x^{2}\).

Step4: Simplify the equation

Subtract \(x^{2}\) from both sides of the equation: \(144=324-36x\).

Step5: Solve for \(x\)

First, rewrite the equation as \(36x=324 - 144\). Then \(36x=180\). Divide both sides by 36: \(x=\frac{180}{36}=5\).

Answer:

\(5\)