Question

similar triangles \\( \triangle a b c \\) and \\( \triangle a d e \\) are graphed on the same coordinate plane.

which statement is true about triangles \\( \triangle a b c \\) and \\( \triangle a d e \\) ?

\\( \bigcirc \\) the triangles have the same area.

\\( \bigcirc \\) the triangles have the same perimeter.

\\( \bigcirc \\) the corresponding sides of the triangles have the same slope.

\\( \bigcirc \\) the corresponding vertices of the triangles share the same coordinate points.

Explanation:

Step1: Analyze the area

Since \(\triangle ABC\) and \(\triangle ADE\) are similar but of different sizes (from the graph), using the formula for the area of a triangle \(A=\frac{1}{2}bh\), they do not have the same area.

Step2: Analyze the perimeter

Using the property of similar triangles (perimeter ratio = side - length ratio), and since they are of different sizes (from the graph), they do not have the same perimeter.

Step3: Analyze the slope

The slope of a line segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For corresponding sides (e.g., \(AB\) and \(AD\), \(AC\) and \(AE\), \(BC\) and \(DE\)), since \(\triangle ABC\sim\triangle ADE\), the rise - over - run (the formula for slope) will be the same. For example, if \(AB\) has endpoints \((x_{A},y_{A})\) and \((x_{B},y_{B})\) and \(AD\) has endpoints \((x_{A},y_{A})\) and \((x_{D},y_{D})\), because of the similarity (proportional sides and same angles), \(\frac{y_{B}-y_{A}}{x_{B}-x_{A}}=\frac{y_{D}-y_{A}}{x_{D}-x_{A}}\)

Step4: Analyze the vertices

From the graph, \(B
eq D\) and \(C
eq E\), so corresponding vertices do not share the same coordinate points.

Answer:

The corresponding sides of the triangles have the same slope.