Questions: If the vertices of triangle DEF are D(-6,3), E(-2,3) and F(0,1), show that triangle DEF is not an isosceles triangle.

Solution Steps

To determine if $\triangle DEF$ is an isosceles triangle, we need to calculate the lengths of its sides using the distance formula. If at least two sides are equal, the triangle is isosceles. Otherwise, it is not.

We will calculate the lengths of the sides of triangle \( \triangle DEF \) using the distance formula:

1. For side \( DE \): \[ DE = \sqrt{((-2) - (-6))^2 + (3 - 3)^2} = \sqrt{(4)^2 + (0)^2} = 4.0 \]

2. For side \( EF \): \[ EF = \sqrt{(0 - (-2))^2 + (1 - 3)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8284 \]

3. For side \( FD \): \[ FD = \sqrt{(0 - (-6))^2 + (1 - 3)^2} = \sqrt{(6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.3246 \]

Now we compare the lengths of the sides:

• \( DE = 4.0 \)
• \( EF \approx 2.8284 \)
• \( FD \approx 6.3246 \)

A triangle is isosceles if at least two of its sides are equal. Here, we see that:

• \( DE \neq EF \)
• \( EF \neq FD \)
• \( FD \neq DE \)

Since none of the sides are equal, \( \triangle DEF \) is not an isosceles triangle.

Final Answer