find the lengths of the altitudes of the triangle whose vertices R, S, and T are given. 𝐑(0,4),

step 1 Today we want the orthocenter, which is where the altitudes cross. And altitudes, remember, need

Find the lengths of the altitudes of the triangle whose...

Key Concepts

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves representing geometric figures using coordinate systems, often the Cartesian coordinate system. It allows for the precise calculation of distances, slopes, and intersections, making it a useful tool for solving problems involving the properties of triangles.

Triangle Altitudes

In a triangle, an altitude is a perpendicular segment drawn from one vertex to the line containing the opposite side. Altitudes are important in determining the area of a triangle and play a key role in various geometric constructions, such as finding the orthocenter, which is the point where the altitudes intersect.

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 fundamental in determining the lengths of sides or segments, including altitudes, between given points.

Slope and Perpendicularity

The slope of a line quantifies its steepness and direction. In coordinate geometry, two lines are perpendicular if the product of their slopes is -1. This relationship is crucial for finding the equation of an altitude, as it must be perpendicular to the side it intersects.

Point-to-Line Distance Formula

The point-to-line distance formula is used to compute the shortest distance from a given point to a specific line in a plane. This formula is particularly useful in finding the length of an altitude from a vertex of a triangle to the opposite side.

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