Question

triangle def is congruent to right triangle abc, shown below. if point f has the coordinates (3,4), what could be the coordinates of point e? (4,-1) (3,-2) (-2,4) (4,-2)

Explanation:

Step1: Determine the lengths of sides in triangle ABC

From the graph, \(A(-2,0)\), \(B(-5,2)\), \(C(-3,-3)\). Calculate \(AB=\sqrt{(-2 + 5)^2+(0 - 2)^2}=\sqrt{9 + 4}=\sqrt{13}\), \(AC=\sqrt{(-2+3)^2+(0 + 3)^2}=\sqrt{1+9}=\sqrt{10}\), \(BC=\sqrt{(-5 + 3)^2+(2 + 3)^2}=\sqrt{4 + 25}=\sqrt{29}\). Since \(\triangle DEF\cong\triangle ABC\), the side - length relationships hold.

Step2: Analyze the distance from point \(F(3,4)\) to each option

• For option \((4,-1)\): Distance \(d=\sqrt{(3 - 4)^2+(4+1)^2}=\sqrt{1 + 25}=\sqrt{26}\)
• For option \((3,-2)\): Distance \(d=\sqrt{(3 - 3)^2+(4 + 2)^2}=6\)
• For option \((-2,4)\): Distance \(d=\sqrt{(3 + 2)^2+(4 - 4)^2}=5\)
• For option \((4,-2)\): Distance \(d=\sqrt{(3 - 4)^2+(4 + 2)^2}=\sqrt{1+36}=\sqrt{37}\). Also, if we consider the right - triangle property (using the grid - based side - length concept, since in a coordinate - plane right - triangle, the vertical and horizontal differences can represent legs). The vertical distance from \(F(3,4)\) to \((4,-2)\) is \(|4-(-2)| = 6\) and the horizontal distance is \(|3 - 4|=1\). In \(\triangle ABC\), if we consider the non - hypotenuse sides (by counting grid units, \(AB\) has a horizontal change of \(3\) and vertical change of \(2\), \(AC\) has a horizontal change of \(1\) and vertical change of \(3\)). Another way: since \(\triangle DEF\cong\triangle ABC\), assume \(DF\) and \(EF\) are the legs. The vertical distance between \(F(3,4)\) and \((4,-2)\) is \(4-(-2)=6\) (equivalent to a side - length in terms of grid units similar to \(BC\)'s non - hypotenuse side concept in the original triangle when considering congruence transformations (translation, rotation, reflection)).

Answer:

\((4,-2)\)