Determine whether the three points are the vertices of a right triangle. (-4,1),(1,4),(-6,-1)

검증된 단계별 안내

Identify the three points given: \(A(-4,1)\), \(B(1,4)\), and \(C(-6,-1)\).

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 lengths of the sides: \(AB\), \(BC\), and \(AC\) by substituting the coordinates into the distance formula.

Check if the triangle is right-angled by verifying the Pythagorean theorem: see if the square of the longest side equals the sum of the squares of the other two sides.

Conclude whether the points form a right triangle based on the Pythagorean theorem check.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Distance Formula

The distance formula calculates the length between two points in the coordinate plane using their coordinates. It is derived from the Pythagorean theorem and is given by √[(x2 - x1)² + (y2 - y1)²]. This formula helps find the lengths of the sides of the triangle formed by the given points.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides. To determine if a triangle is right-angled, check if the side lengths satisfy a² + b² = c², where c is the longest side.

Identifying Right Triangles Using Coordinates

To verify if points form a right triangle, calculate the distances between each pair of points to find side lengths. Then apply the Pythagorean theorem to these lengths. If the relationship holds for any combination, the points form a right triangle.