Question

in the figure below, \\(\overline{az}\\) and \\(\overline{bz}\\) are extensions of line segment \\(\overline{xa}\\) and \\(\overline{yb}\\), which encloses a flowerbed along with sides \\(\overline{xy}\\) and \\(\overline{ab}\\). given \\(\overline{ab} = 20\\), \\(\overline{xy} = 25\\), and \\(\overline{xa} = 6\\) ft, what is the length of \\(\overline{xz}\\), if \\(\angle baz = \angle yxz\\)?

Explanation:

Step1: Prove similarity of triangles

Since \(\angle BAZ=\angle YXZ\) (given) and \(\angle XBA=\angle XZY\) (vertical - angle property), \(\triangle XAB\sim\triangle XYZ\) (by AA - Angle - Angle similarity criterion).

Step2: Set up proportion

For similar triangles \(\triangle XAB\) and \(\triangle XYZ\), the ratios of corresponding sides are equal. That is \(\frac{XA}{XY}=\frac{AB}{YZ}=\frac{XB}{XZ}\). We know \(XA = 6\), \(XY=25\), and \(AB = 20\). Let \(XZ=x\). Then \(\frac{XA}{XY}=\frac{AB}{YZ}\) (not needed for \(XZ\)) and \(\frac{XA}{XY}=\frac{XB}{XZ}\). But also, using the proportion \(\frac{XA}{XY}=\frac{AB}{YZ}\) is not the right path. The correct proportion is \(\frac{AB}{XY}=\frac{XZ - XA}{XZ}\) (since \(XB = XZ - XA\)). Substituting the values: \(\frac{20}{25}=\frac{x - 6}{x}\).
Cross - multiply: \(20x=25(x - 6)\).
Expand: \(20x=25x-150\).

Step3: Solve for \(x\)

Subtract \(20x\) from both sides: \(0 = 25x-20x - 150\).
\(5x=150\).
Divide both sides by \(5\): \(x = 30\).

Answer:

\(30\) ft