Solve each problem. Triangles can be classified by their...

Solve each problem. Triangles can be classified by their sides. (a) An isosceles triangle has at least two sides of equal length. Determine whether the triangle with vertices (0,0),(3,4), and (7,1) is isosceles. (b) An equilateral triangle has all sides of equal length. Determine whether the triangle with vertices (-1,-1), (2,3) , and (-4,3) is equilateral. (c) Determine whether a triangle having vertices (-1,0) (1,0) and (0, \sqrt{3}) is isosceles, equilateral, or neither. (d) Determine whether a triangle having vertices (-3,3) (-2,5) and (-1,3) is isosceles, equilateral, or neither.

Solve each problem.
Triangles can be classified by their sides.
(a) An isosceles triangle has at least two sides of equal length. Determine whether the triangle with vertices (0,0),(3,4), and (7,1) is isosceles.
(b) An equilateral triangle has all sides of equal length. Determine whether the triangle with vertices (-1,-1), (2,3) , and (-4,3) is equilateral.
(c) Determine whether a triangle having vertices (-1,0)
(1,0) and (0, \sqrt{3}) is isosceles, equilateral, or neither.
(d) Determine whether a triangle having vertices (-3,3) (-2,5) and (-1,3) is isosceles, equilateral, or neither.

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Key Concepts

Distance Formula

The distance formula, derived from the Pythagorean theorem, is used to determine the distance between two points in the coordinate plane. It is essential for calculating the lengths of sides of geometric figures, including triangles, based on their vertex coordinates.

Triangle Classification by Side Lengths

Triangles can be classified according to the lengths of their sides. In particular, comparing the computed side lengths of a triangle allows one to determine whether it is scalene (all sides different), isosceles (at least two sides equal), or equilateral (all sides equal).

Isosceles Triangle

An isosceles triangle is defined as a triangle with at least two sides of equal length. Determining if a triangle is isosceles involves calculating the side lengths using the distance formula and comparing them to see if at least two are equal.

Equilateral Triangle

An equilateral triangle is one in which all three sides are of equal length. To verify if a triangle is equilateral, one calculates the distances between all pairs of vertices and checks for equality among all the side lengths.

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