Let A(2,2,-3), B(5,6,9) and C(2,7,9) be the vertices of a triangle. The internal bisector of the angle A meets

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A(2,2,-3)

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Concept understanding — Coordinate Geometry

Coordinate Geometry: Where Algebra Meets Geometry

Imagine you're telling a friend where you left your book in a library. You don't say "near the window" — you say "third shelf, second row, fourth book from the left." You're using numbers to pin down an exact location.

Coordinate geometry does the same thing, but for points on a flat surface. It gives every point a precise address — a pair of numbers — so we can describe shapes, distances, and positions using algebra.

The Big Idea

Before coordinate geometry, geometry was about drawing shapes and proving things with logic alone. Algebra was about numbers and equations. These two worlds seemed separate.

Then René Descartes (a French mathematician) had a simple but revolutionary idea: draw two perpendicular number lines that cross at zero. Now every point on the plane has a unique pair of numbers — its coordinates.

That's it. That's the entire foundation.

The Coordinate System

Take a horizontal line — call it the x-axis. Take a vertical line — call it the y-axis. They cross at a point called the origin, labelled $O$ .

Any point $P$ is located by two numbers:

Its x-coordinate: how far right (positive) or left (negative) from the origin
Its y-coordinate: how far up (positive) or down (negative) from the origin

We write this as an ordered pair: $(x, y)$ .

Note

The order matters. $(3, 5)$ is not the same point as $(5, 3)$ . The first number is always the horizontal position; the second is always the vertical.

A Concrete Example

Plot the point $A(2, 3)$ :

Start at the origin $(0, 0)$ .
Move 2 units to the right along the x-axis.
From there, move 3 units up (parallel to the y-axis).
Mark the point.

Now plot $B(-1, 4)$ :

Start at the origin.
Move 1 unit left (negative x-direction).
Move 4 units up.
Mark the point.

Every point on the plane has exactly one such address. And every pair of numbers corresponds to exactly one point. This one-to-one matching is what makes coordinate geometry powerful.

The Four Quadrants

The axes divide the plane into four regions, called quadrants:

Quadrant	x-sign	y-sign	Example
I	$+$	$+$	$(2, 3)$
II	$-$	$+$	$(-1, 4)$
III	$-$	$-$	$(-3, -2)$
IV	$+$	$-$	$(5, -1)$

Points on the axes themselves (where either coordinate is zero) don't belong to any quadrant.

Why This Matters

Once every point has a number address, we can:

Calculate distances between points using the Pythagorean theorem
Find midpoints by averaging coordinates
Describe lines with equations like $y = mx + c$
Solve geometric problems using algebra instead of drawing

Important

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This is just the Pythagorean theorem in disguise.

The Precise Statement

Coordinate geometry (also called analytic geometry) is the study of geometry using a coordinate system. It establishes a correspondence between:

Points on a plane and ordered pairs of real numbers
Geometric figures (lines, circles, curves) and algebraic equations …

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The internal angle bisector of angle A divides the opposite side BC in the ratio of the adjacent sides AB and AC. By calculating the lengths $AB$ and $AC$ , we find they are equal, meaning D is the midpoint of BC. The coordinates of D are $\boxed{\left(\frac{7}{2}, \frac{13}{2}, 9\right)}$ .

The problem asks for the coordinates of point D, where the internal bisector of angle A of triangle ABC meets side BC. This is a classic application of the Angle Bisector Theorem, which holds true in three-dimensional coordinate geometry just as it does in two dimensions.

The core idea is that an angle bisector divides the opposite side in a ratio equal to the ratio of the other two sides of the triangle. Specifically, if AD is the internal bisector of angle A, then point D divides the side BC in the ratio $AB:AC$ . Once we find this ratio, we can use the section formula to determine the coordinates of D.

Understand the given information and the goal.

We are given the coordinates of the three vertices of a triangle: $A(2,2,-3)$ , $B(5,6,9)$ , and $C(2,7,9)$ . We need to find the coordinates of point D, which lies on BC and is the point where the internal bisector of angle A intersects BC.
Recall the Angle Bisector Theorem.

The Angle Bisector Theorem states that if a line internally bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

In $\triangle ABC$ , if AD is the internal bisector of $\angle A$ , then:

\frac{BD}{DC} = \frac{AB}{AC}

This means that point D divides the line segment BC internally in the ratio $AB:AC$. Let this ratio be $m:n$, where $m = AB$ and $n = AC$.

3. Calculate the lengths of sides AB and AC.

We use the distance formula in 3D for two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ :

> [!FORMULA]

> The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by:

> $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

*   **Length of AB:** Using $A(2,2,-3)$ and $B(5,6,9)$:

AB = \sqrt{(5-2)^2 + (6-2)^2 + (9-(-3))^2}

AB = \sqrt{(3)^2 + (4)^2 + (12)^2}

AB = \sqrt{9 + 16 + 144}

AB = \sqrt{169}

AB = 13

*   **Length of AC:** Using $A(2,2,-3)$ and $C(2,7,9)$:

AC = \sqrt{(2-2)^2 + (7-2)^2 + (9-(-3))^2}

AC = \sqrt{(0)^2 + (5)^2 + (12)^2}

AC = \sqrt{0 + 25 + 144}

AC = \sqrt{169}

AC = 13

Determine the ratio in which D divides BC. From the Angle Bisector Theorem, $D$ divides $BC$ in the ratio $AB:AC$ . We found $AB = 13$ and $AC = 13$ . So, the ratio $m:n = AB:AC = 13:13 = 1:1$ . …

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Q.Let A(2,2,−3)A(2,2,-3)A(2,2,−3), B(5,6,9)B(5,6,9)B(5,6,9) and C(2,7,9)C(2,7,9)C(2,7,9) be the vertices of a triangle. The internal bisector of the angle A meets BC at the point D. Find the coordinates of D.

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Q.Let $A(2,2,-3)$ , $B(5,6,9)$ and $C(2,7,9)$ be the vertices of a triangle. The internal bisector of the angle A meets BC at the point D. Find the coordinates of D.

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