Applications of Distance Formula

List vertices, find needed side lengths, compare lengths, apply the correct property, and write the conclusion with reason.

A rough labelled diagram of the vertices helps decide which pairs of points form sides and diagonals.

Calling a triangle right-angled just because one side is 10 is wrong; the square relation with the largest side must be checked.

Applications of the distance formula use side lengths found from coordinates to identify geometric properties such as equal sides, right triangles, rectangles, or squares.

Applications of the distance formula use side lengths found from coordinates to identify geometric properties such as equal sides, right triangles, rectangles, or squares. Use d = sqrt((x2 - x1)^2 + (y2 - y1)^2), then compare lengths with properties of triangles and quadrilaterals. If AB = 5, BC = 5, and AC = 8, then triangle ABC is isosceles because two sides are equal.

When vertices of a figure are given as coordinates, the distance formula helps find side lengths and sometimes diagonals. After finding lengths, we compare them with known properties: equal sides for isosceles triangle, three equal sides for equilateral triangle, Pythagoras check for right triangle, and equal diagonals with opposite sides equal for rectangles. Use d = sqrt((x2 - x1)^2 + (y2 - y1)^2), then compare lengths with properties of triangles and quadrilaterals. For A(0,0), B(4,0), C(4,3), lengths AB = 4, BC = 3, AC = 5. Since 4^2 + 3^2 = 5^2, triangle ABC is right-angled at B. Questions may ask students to prove that three given points form a right triangle, isosceles triangle, rectangle, square, or rhombus. Calling a triangle right-angled just because one side is 10 is wrong; the square relation with the largest side must be checked.