Determine whether the three points are the vertices of a right triangle. See Example 3. (-6,-4),(0,-2),(-10,8)
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Calculate the distance between each pair of points using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
Find the distance between points (-6, -4) and (0, -2).
Find the distance between points (0, -2) and (-10, 8).
Find the distance between points (-6, -4) and (-10, 8).
Check if the sum of the squares of the two shorter distances equals the square of the longest distance to determine if the triangle is a right triangle (Pythagorean Theorem).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Distance Formula
The distance formula is used to calculate the distance between two points in a coordinate plane. It is given by the formula d = √((x2 - x1)² + (y2 - y1)²). This concept is essential for determining the lengths of the sides of the triangle formed by the three given points.
Solving Quadratic Equations Using The Quadratic Formula
Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is crucial for verifying whether the triangle formed by the three points is a right triangle by checking if the relationship a² + b² = c² holds true.
Collinearity
Collinearity refers to the condition where three or more points lie on a single straight line. In the context of triangles, if the three points are collinear, they do not form a triangle at all. Thus, it is important to first ensure that the points are not collinear before applying the distance formula and Pythagorean theorem.