Interior vs Exterior Math Example 2

Follow the full solution, then compare it with the other examples linked below.

medium

A rectangle has vertices at

(0,0)

(4,0)

(4,3)

, and

(0,3)

. Determine whether the point

(2, 1.5)

is interior, exterior, or on the boundary of the rectangle.

1
Step 1: Identify the boundary of the rectangle. The four sides are: bottom ( $y=0$ , $0 \le x \le 4$ ), top ( $y=3$ , $0 \le x \le 4$ ), left ( $x=0$ , $0 \le y \le 3$ ), right ( $x=4$ , $0 \le y \le 3$ ).
2
Step 2: Check if the point lies on any side. The point is $(2, 1.5)$ . It is not on $y=0$ , $y=3$ , $x=0$ , or $x=4$ , so it is not on the boundary.
3
Step 3: Check if the point is inside. We need $0 < x < 4$ and $0 < y < 3$ . Since $0 < 2 < 4$ and $0 < 1.5 < 3$ , the point satisfies both conditions.
4
Step 4: Conclude that $(2, 1.5)$ is an interior point of the rectangle.

Answer

(2, 1.5)

is an interior point of the rectangle.

For a rectangle aligned with the axes, a point

(x, y)

is interior if its coordinates are strictly between the minimum and maximum

x

- and

y

-values of the rectangle. A point on any edge is on the boundary, and a point outside the rectangle's coordinate ranges is exterior.

About Interior vs Exterior

Interior consists of points strictly inside a boundary; exterior consists of points strictly outside the boundary.

A circle is drawn on a piece of paper. Point [formula] is 3 cm from the center and the radius is 5 c

A triangle is drawn on paper. Point [formula] is inside the triangle and point [formula] is outside.

A point [formula] and a circle centered at [formula] with radius [formula]. Is [formula] interior or