Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, r

Use the diagonals to determine whether a parallelogram with...

Key Concepts

Square Properties

A square combines the properties of both a rectangle and a rhombus; it has congruent sides, equal and perpendicular diagonals, and all right angles. Recognizing both the equal-length and perpendicular characteristics of the diagonals is essential for identifying a square.

Rhombus Properties (Perpendicular Diagonals)

A rhombus features all sides of equal length, and its diagonals intersect at right angles. This perpendicularity of the diagonals, along with the bisecting property, distinguishes a rhombus from other parallelograms.

Distance Formula

The distance formula is used in coordinate geometry to calculate the length of segments, such as the sides or diagonals of a quadrilateral. This tool is fundamental for verifying measurements that determine whether a shape meets the criteria for being a rectangle, rhombus, or square.

Diagonals in Quadrilaterals

The properties of the diagonals, such as whether they are equal in length or perpendicular, are key in distinguishing types of quadrilaterals. In a parallelogram, the diagonals always bisect each other, and additional properties of these diagonals can indicate if the shape is a rectangle, rhombus, or square.

Parallelogram Properties

A parallelogram is defined by its opposite sides being parallel and equal in length, and its diagonals bisect each other. This foundational concept is essential for establishing the shape and relevant properties of any quadrilateral before further classifications are made.

Rectangular Properties (Equal Diagonals)

A rectangle is identified by having all angles as right angles and diagonals that are congruent. The equal-length diagonals are a critical property used to verify that a parallelogram is a rectangle.

Transcript

00:03 Okay, so with this given information, so we wanted to determine if this is a rectangle rhombotaur square.

00:12 So if we start by figuring out these diagonals, so we want to see w .y and x, z.

00:29 Okay, so let's basically see what's happening with each of these.

00:32 So now let's do, so let's do w .y.

00:41 So we want to figure out the length of wy, and to do that, we're going to use the distance formula.

00:48 So we're going to take the difference of the x coordinates, which is going to be 2 minus negative 6 quantity squared, and we're going to add the difference of the y coordinates, which is going to be negative 4 minus 0.

01:12 And we're also going to square that.

01:15 So if we calculate this, so we will get 4 root 5.

01:25 Okay.

01:26 So now let's figure out the length of xz, which is this diagram right here.

01:32 So if we do xz, so we do the same thing, we apply the distance formula.

01:42 So we take the difference of the x coordinates.

01:45 So we're going to do negative 5 minus 1.

01:52 I want to square that, and then we're going to add negative 8 minus 4 squared.

02:05 And i'm going to put that under the square root.

02:08 So now if we solve this, we'll actually get 6 root 5.

02:16 Okay, so notice that 4 root 5 is not equal to 6 root 5.

02:24 So this parallelogram is not a rectangle.