Show that the triangle with vertices A, B and C is a right triangle, and find its area. (GRAPH C

step 1 So in this question, we're given a series of points in the form of a graph. So if we kind of ske

Show that the triangle with vertices A, B and C is a right...

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Key Concepts

Area of a Triangle

The area of a triangle can be calculated using various methods, such as the formula 1/2 * base * height or by using the determinant formula involving the coordinates of its vertices. This concept is essential in problems related to triangles in the coordinate plane where the vertices' positions are known.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. This theorem is frequently applied in coordinate geometry by first calculating distances between points, then checking whether the squared lengths satisfy the theorem.

Right Triangle

A right triangle is one that contains one 90-degree angle. In analytic geometry, demonstrating that a triangle is right often involves showing that two of its sides are perpendicular, such that their slopes multiply to -1, or verifying the angle relationship using the Pythagorean theorem through distance calculations.

Slope and Perpendicularity

The concept of slope is used to describe the steepness of a line in the coordinate plane. For two lines to be perpendicular, the product of their slopes must equal -1, which is a common method for verifying if a triangle contains a right angle when its vertices are given in coordinate form.

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