Question

triangles acd and eba are similar right triangles. the slope of (overline{ad}) is equal to the slope of (overline{ea}). point d has the coordinates ((x, 14)). complete the proportion to show the slope of (overline{ad}=) the slope of (overline{ea}) and to find the exact (x -) coordinate of point d. (\frac{square - 2}{x - 2}=\frac{8}{7}) the exact (x -) coordinate of point d is (square).

Explanation:

Step1: Find the numerator of the left - hand side fraction

The \(y\) - coordinate of point \(D\) is \(14\) and the \(y\) - coordinate of point \(A\) is \(2\). So the numerator of the left - hand side fraction (which represents the change in \(y\) for segment \(AD\)) is \(14 - 2\).

Step2: Set up the proportion using the slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For segment \(EA\), \(y_E = 10\), \(y_A=2\), \(x_E = 8\), \(x_A = 2\). The slope of \(EA\) is \(\frac{10 - 2}{8 - 2}=\frac{8}{6}=\frac{4}{3}\). For segment \(AD\), the slope is \(\frac{14 - 2}{x - 2}\). Since the slope of \(\overline{AD}\) equals the slope of \(\overline{EA}\), we have \(\frac{14 - 2}{x - 2}=\frac{8}{6}\).
Cross - multiply: \(6\times(14 - 2)=8\times(x - 2)\)
\(6\times12 = 8\times(x - 2)\)
\(72=8x-16\)
Add \(16\) to both sides: \(72 + 16=8x\)
\(88 = 8x\)
Divide both sides by \(8\): \(x = 11\)

Answer:

The first box is \(14\), and the \(x\) - coordinate of point \(D\) is \(11\).