Question

The vertices of △DEF are D(7, 3), E(4, -3), and F(10, -3). Write a paragraph proof to prove that △DEF is isosceles.

Answer 1

By calculating the distances between the vertices of triangle DEF using the distance formula, we established that sides DE and DF are of equal length, hence triangle DEF is an isosceles triangle.

Step-by-step explanation:

To prove that triangle DEF is isosceles, we need to show that at least two sides of the triangle have equal lengths. Using the coordinates of the vertices D(7, 3), E(4, -3), and F(10, -3), we can calculate the lengths of the sides using the distance formula: Distance = √[(x2 - x1)^2 + (y2 - y1)^2].

For side DE, we have:

• Distance DE = √[(4 - 7)^2 + (-3 - 3)^2] = √[9 + 36] = √[45]

For side DF, we have:

• Distance DF = √[(10 - 7)^2 + (-3 - 3)^2] = √[9 + 36] = √[45]

For side EF, we judge by the coordinates E(4, -3) and F(10, -3) that they share the same y-coordinate, which means they are horizontally aligned and we can measure the distance directly:

• Distance EF = |10 - 4| = 6

As the calculations show, distance DE is equal to distance DF, which means that triangle DEF has two sides of equal length, proving that it is indeed an isosceles triangle.