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

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

Key Concepts

Distance Formula

The distance formula is used to calculate the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is essential for finding the lengths of the sides of a triangle when the coordinates of the vertices are known. This helps in verifying the presence of a right angle using the Pythagorean theorem and in computing the triangle’s area.

Area of a Triangle

The area of a triangle can be calculated using various methods, including the traditional formula of one-half the product of its base and height or using the coordinates of its vertices through formulas like the shoelace formula. Understanding how to compute the area of a triangle is key to solving problems where the dimensions or positions of the triangle are provided in a coordinate system.

Slope of a Line

The slope of a line in a coordinate plane measures its steepness, typically calculated by the ratio of the change in the y-coordinate to the change in the x-coordinate between two points. Determining the slopes of the sides of a triangle helps in assessing whether any pair of sides are perpendicular, as the slopes of perpendicular lines are negative reciprocals of each other.

Right Triangle

A right triangle is a type of triangle that has one 90-degree angle. In coordinate geometry, one way to identify a right angle is by checking if the slopes of two sides are negative reciprocals of each other, which indicates that the sides are perpendicular. Alternatively, one can use the Pythagorean theorem to verify that the sum of the squares of the lengths of the two shorter sides equals the square of the length of the longest side.

Perpendicular Lines

Two lines are perpendicular if they intersect to form a right angle, and in a coordinate plane, this occurs when the product of their slopes is -1. This concept is crucial in determining whether a triangle has a right angle by comparing the slopes of its adjacent sides.

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