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6.4: Rectangles, Rhombuses and Squares

Created by: CK-12

Learning Objectives

  • Define a rectangle, rhombus, and square.
  • Determine if a parallelogram is a rectangle, rhombus, or square in the x-y plane.
  • Compare the diagonals of a rectangle, rhombus, and square.

Review Queue

  1. List five examples where you might see a square, rectangle, or rhombus in real life.
  2. Find the values of x and y that would make the quadrilateral a parallelogram.

Know What? You are designing a patio for your backyard and are marking it off with a tape measure. Two sides are 21 feet long and two sides are 28 feet long. Explain how you would only use the tape measure to make your patio a rectangle. (You do not need to find any measurements.)

Defining Special Parallelograms

Rectangles, Rhombuses (also called Rhombi) and Squares are all more specific versions of parallelograms, also called special parallelograms. Taking the theorems we learned in the previous two sections, we have three more new theorems.

Rectangle Theorem: A quadrilateral is a rectangle if and only if it has four right (congruent) angles.

ABCD is a rectangle if and only if \angle A \cong \angle B \cong \angle C \cong \angle D.

Rhombus Theorem: A quadrilateral is a rhombus if and only if it has four congruent sides.

ABCD is a rhombus if and only if \overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{AD}.

Square Theorem: A quadrilateral is a square if and only if it has four right angles and four congruent sides.

ABCD is a square if and only if \angle A \cong \angle B \cong \angle C \cong \angle D and \overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{AD}.

From the Square Theorem, we can also conclude that a square is a rectangle and a rhombus.

Example 1: What type of parallelogram are the ones below?

a)

b)

Solution:

a) All sides are congruent and one angle is 135^\circ, so the angles are not congruent. This is a rhombus.

b) All four congruent angles and the sides are not. This is a rectangle.

Example 2: Is a rhombus SOMETIMES, ALWAYS, or NEVER a square? Explain why.

Solution: A rhombus has four congruent sides and a square has four congruent sides and angles. Therefore, a rhombus is a square when it has congruent angles. This means a rhombus is SOMETIMES a square.

Example 3: Is a rectangle SOMETIMES, ALWAYS, or NEVER a parallelogram? Explain why.

Solution: A rectangle has two sets of parallel sides, so it is ALWAYS a parallelogram.

Diagonals in Special Parallelograms

Recall from previous lessons that the diagonals in a parallelogram bisect each other. Therefore, the diagonals of a rectangle, square and rhombus also bisect each other. They also have additional properties.

Investigation 6-3: Drawing a Rectangle

Tools Needed: pencil, paper, protractor, ruler

  1. Like with Investigation 6-2, draw two lines on either side of your ruler, making them parallel. Make these lines 3 inches long.
  2. Using the protractor, mark two 90^\circ angles, 2.5 inches apart on the bottom line from Step 1. Extend the sides to intersect the top line.
  3. Draw in the diagonals and measure. What do you discover?

Theorem 6-14: A parallelogram is a rectangle if the diagonals are congruent.

ABCD is parallelogram. If \overline{AC} \cong \overline{BD}, then ABCD is also a rectangle.

Investigation 6-4: Drawing a Rhombus

Tools Needed: pencil, paper, protractor, ruler

1. Like with Investigation 6-2 and 6-3, draw two lines on either side of your ruler, 3 inches long.

2. Remove the ruler and mark a 50^\circ angle, at the left end of the bottom line drawn in Step 1. Draw the other side of the angle and make sure it intersects the top line. Measure the length of this side.

3. Mark the length found in Step 2 on the bottom line and the top line from the point of intersection with the 50^\circ angle. Draw in the fourth side. It will connect the two endpoints of these lengths.

4. By the way we drew this parallelogram; it is a rhombus because all the sides are equal. Draw in the diagonals.

Measure the angles at the point of intersection of the diagonals (4).

Measure the angles created by the sides and each diagonal (8).

Theorem 6-15: A parallelogram is a rhombus if the diagonals are perpendicular.

ABCD is a parallelogram. If \overline{AC} \perp \overline{BD}, then ABCD is also a rhombus.

Theorem 6-16: A parallelogram is a rhombus if the diagonals bisect each angle.

ABCD is a parallelogram. If \overline{AC} bisects \angle BAD and \angle BCD and \overline{BD} bisects \angle ABC \ \angle ADC, then ABCD is also a rhombus.

The converses of these three theorems are true. There are no theorems about the diagonals of a square. The diagonals of a square have the properties of a rhombus and a rectangle.

Example 4: List everything you know about the square SQRE.

Solution: A square has all the properties of a parallelogram, rectangle and rhombus.

Properties of a Parallelogram Properties of a Rhombus Properties of a Rectangle
  • \overline{SQ} \| \overline{ER}
  • \overline{SQ} \cong \overline{ER} \cong \overline{SE} \cong \overline{QR}
  • m \angle SER = m \angle SQR = m \angle QSE = m \angle QRE  = 90^\circ
  • \overline{SE} \| \overline{QR}
  • \overline{SR} \perp \overline{QE}
  •  \angle SEQ \cong \angle QER \cong \angle SQE \cong \angle EQR
  • \overline{SR} \cong \overline{QE}
  • \angle QSR \cong \angle RSE \cong \angle QRS \cong \angle SRE
  • \overline{SA} \cong \overline{AR} \cong \overline{QA} \cong \overline{AE}

All the bisected angles are 45^\circ.

Parallelograms in the Coordinate Plane

Example 4: Determine what type of parallelogram TUNE is: T(0, 10), U(4, 2), N(-2, -1), and E(-6, 7).

Solution: Let’s see if the diagonals are equal. If they are, then TUNE is a rectangle.

EU & = \sqrt{(-6 -4)^2 + (7-2)^2} &&	TN = \sqrt{(0 + 2)^2 +(10 + 1)^2}\\& = \sqrt{(-10)^2 + 5^2} && \quad \ \ = \sqrt{2^2 + 11^2}\\& = \sqrt{100 + 25} && \quad \ \ = \sqrt{4 + 121}\\& = \sqrt{125} && \quad \ \ = \sqrt{125}

If the diagonals are also perpendicular, then TUNE is a square.

\text{Slope of}\ EU = \frac{7 - 2}{-6 - 4} = -\frac{5}{10} = -\frac{1}{2} \quad \text{Slope of}\ TN = \frac{10 - 7}{0-(-6)} = \frac{3}{6} = \frac{1}{2}

The slope of EU \neq slope of TN, so TUNE is a rectangle.

Steps to determine if a quadrilateral is a parallelogram, rectangle, rhombus, or square.

1. Graph the four points on graph paper.

2. See if the diagonals bisect each other. (midpoint formula)

Yes: Parallelogram, continue to #2. No: A quadrilateral, done.

3. See if the diagonals are equal. (distance formula)

Yes: Rectangle, skip to #4. No: Could be a rhombus, continue to #3.

4. See if the sides are congruent. (distance formula)

Yes: Rhombus, done. No: Parallelogram, done.

5. See if the diagonals are perpendicular. (find slopes)

Yes: Square, done. No: Rectangle, done.

Know What? Revisited In order for the patio to be a rectangle, the opposite sides must be congruent (see picture). To ensure that the parallelogram is a rectangle without measuring the angles, the diagonals must be equal.

Review Questions

  • Questions 1-3 are similar to #2 in the Review Queue and Example 1.
  • Questions 4-15 are similar to Example 1.
  • Questions 16-21 are similar to Examples 2 and 3.
  • Questions 22-25 are similar to Investigations 6-3 and 6-4.
  • Questions 26-29 are similar to Example 4.
  • Question 30 is a challenge.
  1. RACE is a rectangle. Find:
    1. RG
    2. AE
    3. AC
    4. EC
    5. m \angle RAC

  2. DIAM is a rhombus. Find:
    1. MA
    2. MI
    3. DA
    4. m \angle DIA
    5. m \angle MOA

  3. CUBE is a square. Find:
    1. m \angle UCE
    2. m \angle EYB
    3. m \angle UBY
    4. m \angle UEB

For questions 4-15, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none.

For questions 16-21 determine if the following are ALWAYS, SOMETIME, or NEVER true. Explain your reasoning.

  1. A rectangle is a rhombus.
  2. A square is a parallelogram.
  3. A parallelogram is regular.
  4. A square is a rectangle.
  5. A rhombus is equiangular.
  6. A quadrilateral is a pentagon.

Construction Draw or construct the following quadrilaterals.

  1. A quadrilateral with congruent diagonals that is not a rectangle.
  2. A quadrilateral with perpendicular diagonals that is not a rhombus or square.
  3. A rhombus with a 6 cm diagonal and an 8 cm diagonal.
  4. A square with 2 inch sides.

For questions 26-29, determine what type of quadrilateral ABCD is. Use Example 4 and the steps following it to help you.

  1. A(-2, 4), B(-1, 2), C(-3, 1), D(-4, 3)
  2. A(-2, 3), B(3, 4), C(2, -1), D(-3, -2)
  3. A(1, -1), B(7, 1), C(8, -2), D(2, -4)
  4. A(10, 4), B(8, -2), C(2, 2), D(4, 8)
  5. Challenge SRUE is a rectangle and PRUC is a square.
    1. What type of quadrilateral is SPCE?
    2. If SR = 20 and RU = 12, find CE.
    3. Find SC and RC based on the information from part b. Round your answers to the nearest hundredth.

Review Queue Answers

  1. Possibilities: picture frame, door, baseball diamond, windows, walls, floor tiles, book cover, pages/paper, table/desk top, black/white board, the diamond suit (in a deck of cards).
    1. x = 11, \ y = 6
    2. x = y = 90^\circ
    3. x = 9, \ y = 133^\circ

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Grades:

8 , 9 , 10

Date Created:

Feb 22, 2012

Last Modified:

Jul 10, 2014
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CK.MAT.ENG.SE.1.Geometry-Basic.6.4

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