Question

triangles abc and triangle cde are similar right triangles. complete the proportion to show that the slopes of both of the sides of the triangles that lie on line ae are equal. enter a number in each of the boxes.
\\( \frac{1.5}{2}=\frac{\square}{\square} \\)

Explanation:

Step1: Recall the slope formula

The slope \(m\) of a line segment with vertical change (rise) \(y\) and horizontal change (run) \(x\) is \(m=\frac{\text{rise}}{\text{run}}\). For similar right - triangles, the ratios of their corresponding sides (rise over run) are equal.

Step2: Identify the values for the first triangle

For triangle \(ABC\), the vertical side (rise) \(BC = 1.5\) and the horizontal side (run) \(AB=2\).

Step3: Identify the values for the second triangle

For triangle \(CDE\), the vertical side (rise) \(DE\) and the horizontal side (run) \(CD = 4\). Since the triangles are similar, \(\frac{BC}{AB}=\frac{DE}{CD}\).
We know that the ratio of the sides of similar triangles is constant. The slope of the line segment in triangle \(ABC\) is \(\frac{1.5}{2}\), and for triangle \(CDE\), let the vertical side be \(y\).
Since the triangles are similar, \(\frac{1.5}{2}=\frac{y}{4}\). Cross - multiplying gives \(2y=1.5\times4\), so \(y = 3\).

Answer:

\(\frac{1.5}{2}=\frac{3}{4}\)