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To classify the triangle by its sides, we need to calculate the distances between each pair of points, which will give us the lengths of the sides of the triangle. The points are \(D(-2, 3)\), \(E(5, 5)\), and \(F(-4, 10)\).

The distance formula is:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

**1. Distance \(DE\):**

\[
DE = \sqrt{(5 + 2)^2 + (5 - 3)^2} = \sqrt{7^2 + 2^2} = \sqrt{49 + 4} = \sqrt{53}
\]

**2. Distance \(EF\):**

\[
EF = \sqrt{(-4 - 5)^2 + (10 - 5)^2} = \sqrt{(-9)^2 + 5^2} = \sqrt{81 + 25} = \sqrt{106}
\]

**3. Distance \(FD\):**

\[
FD = \sqrt{(-4 + 2)^2 + (10 - 3)^2} = \sqrt{(-2)^2 + 7^2} = \sqrt{4 + 49} = \sqrt{53}
\]

Since \(DE = FD = \sqrt{53}\) and \(EF = \sqrt{106}\), the triangle has two sides of equal length.

Therefore, the triangle is **isosceles**.

Learn how to classify a triangle by its sides using the distance formula. The triangle is formed by points D(-2, 3), E(5, 5), and F(-4, 10). Includes step-by-step calculations and the identification of the triangle as isosceles. Suitable for Grades 8-10.

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