PQRS is a rectangle. a Find the coordinates of point P. b Find the area of the rectangle. (FIGUR

step 1 So here we have a rectangle drawn on the coordinate plane. You can see the X and Y axis there. A

PQRS is a rectangle. a Find the coordinates of point P. b...

Key Concepts

Rectangle Properties

A rectangle is a quadrilateral with four right angles, where opposite sides are parallel and congruent. One of its important properties is that its diagonals are equal in length and bisect each other, which can be very useful in determining unknown vertex coordinates.

Coordinate Geometry

This field involves using a coordinate system to describe geometric figures. It includes the use of formulas for distance, slope, and midpoint, which are essential for analyzing and solving problems related to the placement and properties of shapes in the plane.

Area Calculation of a Rectangle

The area of a rectangle is calculated by multiplying the lengths of any two adjacent sides. In coordinate geometry, these side lengths can be determined by computing the differences between the corresponding coordinates of the vertices.

Use of Diagonals to Find Missing Coordinates

When some coordinates of a rectangle are known, the property that the diagonals bisect each other can help locate missing points. By finding the midpoint of one diagonal, you can set it equal to the midpoint of the other, creating a system of equations to solve for the unknown coordinates.

Transcript

00:03 So here we have a rectangle drawn on the coordinate plane.

00:07 You can see the x and y axis there.

00:09 And we're being asked to determine the coordinates of point p as well as the area of that rectangle.

00:20 Now, because the rectangle is situated in such a way that two of the sides are parallel to the x axis and two of the sides are parallel to the y axis, the coordinates of the points are very easy to identify.

00:37 You see here, those two x coordinates are the same because that side, sr, or rs, is a vertical segment.

00:48 So that means this x coordinate and this x coordinate for point p must also be the same because a vertical segment, the x coordinates will be the same.

01:03 So that tells me the x coordinate of point p is negative three.

01:07 And now if you look at the y coordinates, let me change my highlight color here.

01:13 You'll notice same thing except in a horizontal orientation.

01:18 That y coordinate and that y coordinate will be the same because that segment is horizontal.

01:27 And all horizontal lines or horizontal segments have the same y coordinate.

01:33 That's why the equation of a horizontal line is simply y equals some number.

01:40 So using that same logic, we know that that y coordinate and that y coordinate have to be the same.

01:52 So that tells us the coordinates of point p are negative 3, 2.

02:03 Part b asks us to find the area of that rectangle.

02:09 Well, this is very easy because the segments, again, the side links, i should say the line, line segments that make up each side, you can very easily determine their length because they're horizontal and vertical.

02:24 So, of course, the area of a rectangle is length times width.

02:30 Let's see if we can find those.

02:33 So for this side right here, let me highlight it, let's call that the length...

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