rectangle

Rectangle: Shape and Properties

A special kind of parallelogram

A rectangle is a parallelogram with 4 right angles.  Now, since a rectangle is a parallelogram, its opposite sides must be congruent and it must satisfy all other properties of parallelograms

The Properties of a Rectangle

4 Right Angles

Rectangle angles

In a rectangle, all angles are 90°

Diagonals of Rectangle

Rectangle angles

The diagonals of a rectangle are congruent.

It's easy to prove that the diagonals of a rectangle with the Pythagorean theorem. Click here to see the proof.

Practice Problems

Problem 1

In rectangle STAR below, SA =5, what is the length of RT?

Sides of Rectangle

Since the diagonals of a rectangle are congruent, RT has the same length as SA.

Therefore, RT = 5

Problem 2

If side MN = 12 and side ML = 5, what is the length of the other two sides?

Sides of Rectangle

Side LO = 12 and NO = 5

Remember that a rectangle is a parallelogram, so it has all of the properties of parallelograms , including congruent opposite sides.

Problem 3

How Long is MO and MZ in the rectangle pictured on the left?

Rectangle Practice Problem
Finding length of MO

Since the diagonals of a rectangle are congruent MO = 26.

Finding length of MZ

To find MZ, you must  remember that the diagonals of a parallelogram bisect each other.(Remember a rectangle is a type of parallelogram so rectangles get all of the parallelogram properties)

If MO = 26 and the diagonals bisect each other, then MZ = ½(26) = 13

Challenge Problem

What is the value of x in rectangle STAR below?

challenge problem

$$ \angle SZT $$ and $$ \angle SZA $$ are supplementary angles,
Therefore $$ \angle SZA = 120° $$

Next, remember that the diagonals of any parallelogram bisect each other and the diagonals of a rectangle are congruent.

Therefore, SZ = AZ, making SZA isosceles and $$ \angle $$ZSAcongruent$$ \angle $$ZAS, being base angles of an isosceles triangle. Therefore, x = 30 °

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