A(1,-3,3 / 2), B(3,-7,4), C(p,-3, q) are the vertices of a ABC. If the median through A is equal

step 1 So in this case we need to find the value of p and q. So in this case, variant through A is equa

A(1,-3,3 / 2), B(3,-7,4), C(p,-3, q) are the vertices of a...

$\mathrm{A}(1,-3,3 / 2), \mathrm{B}(3,-7,4), \mathrm{C}(\mathrm{p},-3, \mathrm{q})$ are the vertices of a $\triangle \mathrm{ABC}$. If the median through $\mathrm{A}$ is equally inclined to the positive directions of coordinate axes, then the values of $\mathrm{p}$ and $\mathrm{q}$ are respectively. (a) $-5,-5$ (b) $-5,5$ (c) $5,-5$ (d) 5,5

Key Concepts

Median of a Triangle in 3D

In three?dimensional geometry, the median of a triangle is the line segment joining one vertex to the midpoint of the opposite side. This concept is crucial because it provides a means to analyze the distribution of a triangle’s vertices in space and is often used to derive further geometric properties.

Midpoint Formula

The midpoint formula in 3D is a fundamental concept in coordinate geometry, where the midpoint of a segment between two points is calculated by taking the average of the corresponding coordinates of those points. This is used to determine the exact center of the side opposite to the given vertex for constructing the median.

Direction Cosines and Equal Inclination

Direction cosines correspond to the cosines of the angles that a vector makes with the positive directions of the coordinate axes. When a line is equally inclined to all three axes, its direction cosines are identical. This property is used to establish relationships among the vector components of the median, leading to conditions that must be satisfied by unknown coordinates.

Vector Components in 3D Geometry

In the context of 3D coordinate geometry, vectors represent directions and magnitudes. Analyzing the components of a vector, such as that defining a median, plays a crucial role in problems where relationships between geometric entities are set by specific angle conditions, such as equal inclination to the coordinate axes.

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