Parallelogram - Parallelogram Verification

Did you notice the quadrilateral that is formed at the intersection of 2 train tracks? What is it called? What are its characteristics? Let's take a look at the train tracks, why are train tracks 2 parallel tracks? For the train to not derail, there must be 2 tracks that always maintain the same distance apart. This is the definition of parallel lines that never meet because the distance between them is always equal. This is the definition of parallel lines that never meet because the distance between them is always equal. At the moment when 2 train tracks meet, a quadrilateral is formed between them, which has 2 pairs of opposite sides parallel, which is the parallelogram

If the data is:

  • ABǁCD AB ǁ CD
  • ADǁBC AD ǁ BC

Then: ABCD ABCD is a parallelogram

Parallelogram

A1 - Parallelogram KLMN

Practice Parallelogram

Exercise #1

Below is a quadrilateral:

Is it possible that it is a parallelogram?

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Step-by-Step Solution

According to the properties of the parallelogram: the diagonals intersect each other.

From the data in the drawing, it follows that diagonal AC and diagonal BD are divided into two equal parts, that is, the diagonals intersect each other:

AO=OC=8 AO=OC=8

DO=OB=10 DO=OB=10

Therefore, the quadrilateral is actually a parallelogram.

Answer

Yes

Exercise #2

Below is a quadrilateral:

Is it possible that it is a parallelogram?

AAABBBCCCDDD1206012060

Step-by-Step Solution

Let's review the property: a quadrilateral in which two pairs of opposite angles are equal is a parallelogram.

From the data in the drawing, it follows that:

D=B=60 D=B=60

A=C=120 A=C=120

Therefore, the quadrilateral is actually a parallelogram.

Answer

Yes

Exercise #3

Below is a quadrilateral:

Is it possible that it is a parallelogram?

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Step-by-Step Solution

According to the properties of a parallelogram, any two opposite sides will be equal to each other.

From the data, it can be observed that only one pair of opposite sides are equal and therefore the quadrilateral is not a parallelogram.

Answer

No

Exercise #4

Look at the parallelogram in the figure.

Its area is equal to 70 cm².

Calculate DC.

555AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

The formula for the area of a parallelogram:

Height * The side to which the height descends.

We replace in the formula all the known data, including the area:

5*DC = 70

We divide by 5:

DC = 70/5 = 14

And that's how we reveal the unknown!

Answer

14 14 cm

Exercise #5

AB = DC.=

Is the shape below a parallelogram?

AAABBBCCCDDDx+582x+9x

Step-by-Step Solution

In a parallelogram, we know that each pair of opposite sides are equal to each other.

The data shows that only one pair of sides are equal to each other:

AB=DC=8 AB=DC=8

Now we try to see that the additional pair of sides are equal to each other.

We replacex=8 x=8 for each of the sides:

AD=2×8+9 AD=2\times8+9

AD=16+9 AD=16+9

AD=25 AD=25

BC=8+5 BC=8+5

BC=13 BC=13

That is, we find that the pair of opposite sides are not equal to each other:

2513 25\ne13

Therefore, the quadrilateral is not a parallelogram.

Answer

No

Exercise #1

Below is a quadrilateral:

Given B+C=180 ∢B+∢C=180

Is it possible that it is a parallelogram?

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Step-by-Step Solution

Remember that in a parallelogram each pair of opposite angles are equal to each other.

The data shows that only one pair of angles are equal to each other:

D=B=140 D=B=140

Therefore, we will now find angle C and see if it is equal to angle A, that is, if angle C is equal to 40:

Let's remember that a pair of angles on the same side are equal to 180 degrees, therefore:

B+C=180 B+C=180

We replace the existing data:

140+4x=180 140+4x=180

4x=180140 4x=180-140

4x=40 4x=40

Divide by 4:

4x4=404 \frac{4x}{4}=\frac{40}{4}

x=10 x=10

Now we replace X:

C=4×10=40 C=4\times10=40

That is, we found that angles A and C are equal to each other and that the quadrilateral is a parallelogram since each pair of opposite angles are equal to each other.

Answer

Yes

Exercise #2

Look at the parallelogram in the figure below.

Its area is equal to 40 cm².

Calculate AE.

888AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Given that ABCD is a parallelogram,AB=CD=8 AB=CD=8 According to the properties of the parallelogram, each pair of opposite sides are equal and parallel.

To find AE we will use the area given to us in the formula to find the area of the parallelogram:

S=DC×AE S=DC\times AE

40=8×AE 40=8\times AE

We divide both sides of the equation by 8:

8AE:8=40:8 8AE:8=40:8

AE=5 AE=5

Answer

5 5 cm

Exercise #3

Look at the quadrilateral below.

AO = OC

Is it a parallelogram?

AAABBBCCCDDDOOO5x+49x+110x3x-2

Step-by-Step Solution

Let's pay attention to the diagonals, remember that in a parallelogram the diagonals intersect each other.

Therefore, we will find AO, OC, BO, DO and check if they are equal and intersect each other.

We refer to the figure:

AO=OC AO=OC

9x+1=10x 9x+1=10x

We place like terms:

1=10x9x 1=10x-9x

1=x 1=x

We replace:

AO=9×1+1=10 AO=9\times1+1=10

OC=10×1=10 OC=10\times1=10

Now we know that indeedAO=OC AO=OC

Now we establish that X=1 and see if BO is equal to OD:

BO=3x2 BO=3x-2

BO=3×12= BO=3\times1-2=

BO=32=1 BO=3-2=1

OD=5x+4 OD=5x+4

OD=5×1+4 OD=5\times1+4

OD=5+4=9 OD=5+4=9

Now we find that: BOOD BO\ne OD

Since the diagonals do not intersect each other, the quadrilateral is not a parallelogram.

Answer

No

Exercise #4

ABCD parallelogram, it is known that:

BE is perpendicular to DE

BF is perpendicular to DF

BF=8 BE=4 AD=6 DC=12

Calculate the area of the parallelogram in 2 different ways

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Video Solution

Step-by-Step Solution

In this exercise, we are given two heights and two sides.

It is important to keep in mind: The external height can also be used to calculate the area

Therefore, we can perform the operation of the following exercise:

The height BF * the side AD

8*6

 

The height BE the side DC
4
*12

 The solution of these two exercises is 48, which is the area of the parallelogram.

 

Answer

48 cm²

Exercise #5

ABCD is a parallelogram.

CE is its height.

CB = 5
AE = 7
EB = 2

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What is the area of the parallelogram?

Video Solution

Step-by-Step Solution

To find the area,

first, the height of the parallelogram must be found.

To conclude, let's take a look at triangle EBC.

Since we know it is a right triangle (since it is the height of the parallelogram)

the Pythagorean theorem can be used:

a2+b2=c2 a^2+b^2=c^2

In this case: EB2+EC2=BC2 EB^2+EC^2=BC^2

We place the given information: 22+EC2=52 2^2+EC^2=5^2

We isolate the variable:EC2=52+22 EC^2=5^2+2^2

We solve:EC2=254=21 EC^2=25-4=21

EC=21 EC=\sqrt{21}

Now all that remains is to calculate the area.

It is important to remember that for this, the length of each side must be used.
That is, AE+EB=2+7=9

21×9=41.24 \sqrt{21}\times9=41.24

Answer

41.24

Exercise #1

AE is the height of the parallelogram ABCD.

AB is 3 cm longer than AE.

The area of ABCD is 32 cm².

Calculate the length of side AB.

S=32S=32S=32AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Keep in mind that AB is 3 cm greater than AE, so we must pay attention to the data when we put the formula to calculate the parallelogram:

Height multiplied by the side of the height:

AB×AE=S AB\times AE=S

We will mark AE with the letter a and therefore AB will be a+3:

a×(a+3)=32 a\times(a+3)=32

We open the parentheses:

a2+3a=32 a^2+3a=32

We use the trinomial/roots formula:

a2+3a32=0 a^2+3a-32=0 (a+8)(a5)=0 (a+8)(a-5)=0

That means we have two options:

a=8,a=5 a=-8,a=5

Since it is not possible to place a negative side in the formula to calculate the areaa=5 a=5

Now we can calculate the sides:

AE=5 AE=5

AB=5+3=8 AB=5+3=8

Answer

8 cm

Exercise #2

The parallelogram ABCD contains the rectangle AEFC inside it, which has a perimeter of 24.

AE = 8

BC = 5

P=24P=24P=24555AAABBBCCCDDDEEEFFF8

What is the area of the parallelogram?

Video Solution

Step-by-Step Solution

In the first step, we must find the length of EC, which we will identify with an X.

We know that the perimeter of a rectangle is the sum of all its sides (AE+EC+CF+FA),

Since in a rectangle the opposite sides are equal, the formula can also be written like this: 2AE=2EC.

We replace the known data:

2×8+2X=24 2\times8+2X=24

16+2X=24 16+2X=24

We isolate X:

2X=8 2X=8

and divide by 2:

X=4 X=4

Now we can use the Pythagorean theorem to find EB.

(Pythagoras: A2+B2=C2 A^2+B^2=C^2 )

EB2+42=52 EB^2+4^2=5^2

EB2+16=25 EB^2+16=25

We isolate the variable

EB2=9 EB^2=9

We take the square root of the equation.

EB=3 EB=3

The area of a parallelogram is the height multiplied by the side to which the height descends, that isAB×EC AB\times EC .

AB= AE+EB AB=\text{ AE}+EB

AB=8+3=11 AB=8+3=11

And therefore we will apply the area formula:

11×4=44 11\times4=44

Answer

44

Exercise #3

The area of trapezoid ABCD is X cm².

The line AE creates triangle AED and parallelogram ABCE.

The ratio between the area of triangle AED and the area of parallelogram ABCE is 1:3.

Calculate the ratio between sides DE and EC.

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Video Solution

Step-by-Step Solution

To calculate the ratio between the sides we will use the existing figure:

AAEDAABCE=13 \frac{A_{AED}}{A_{ABCE}}=\frac{1}{3}

We calculate the ratio between the sides according to the formula to find the area and then replace the data.

We know that the area of triangle ADE is equal to:

AADE=h×DE2 A_{ADE}=\frac{h\times DE}{2}

We know that the area of the parallelogram is equal to:

AABCD=h×EC A_{ABCD}=h\times EC

We replace the data in the formula given by the ratio between the areas:

12h×DEh×EC=13 \frac{\frac{1}{2}h\times DE}{h\times EC}=\frac{1}{3}

We solve by cross multiplying and obtain the formula:

h×EC=3(12h×DE) h\times EC=3(\frac{1}{2}h\times DE)

We open the parentheses accordingly:

h×EC=1.5h×DE h\times EC=1.5h\times DE

We divide both sides by h:

EC=1.5h×DEh EC=\frac{1.5h\times DE}{h}

We simplify to h:

EC=1.5DE EC=1.5DE

Therefore, the ratio between is: ECDE=11.5 \frac{EC}{DE}=\frac{1}{1.5}

Answer

1:1.5 1:1.5

Exercise #4

ABCD is a parallelogram with a perimeter of 38 cm.

AB is twice as long as CE.

AD is three times shorter than CE.

CE is the height of the parallelogram.

Calculate the area of the parallelogram.

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Video Solution

Step-by-Step Solution

Let's call CE as X

According to the data

AB=x+2,AD=x3 AB=x+2,AD=x-3

The perimeter of the parallelogram:

2(AB+AD) 2(AB+AD)

38=2(x+2+x3) 38=2(x+2+x-3)

38=2(2x1) 38=2(2x-1)

38=4x2 38=4x-2

38+2=4x 38+2=4x

40=4x 40=4x

x=10 x=10

Now it can be argued:

AD=103=7,CE=10 AD=10-3=7,CE=10

The area of the parallelogram:

CE×AD=10×7=70 CE\times AD=10\times7=70

Answer

70 cm²

Exercise #5

ABCD is a parallelogram whose perimeter is equal to 24 cm.

The side of the parallelogram is two times greater than the adjacent side (AB>AD).

CE is the height of the side AB

The area of the parallelogram is 24 cm².

Find the height of CE

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Video Solution

Step-by-Step Solution

The perimeter of the parallelogram is calculated as follows:

SABCD=AB+BC+CD+DA S_{ABCD}=AB+BC+CD+DA Since ABCD is a parallelogram, each pair of opposite sides is equal, and therefore, AB=DC and AD=BC

According to the figure that the side of the parallelogram is 2 times larger than the side adjacent to it, it can be argued thatAB=DC=2BC AB=DC=2BC

We inut the data we know in the formula to calculate the perimeter:

PABCD=2BC+BC+2BC+BC P_{ABCD}=2BC+BC+2BC+BC

We replace the given perimeter in the formula and add up all the BC coefficients accordingly:

24=6BC 24=6BC

We divide the two sections by 6

24:6=6BC:6 24:6=6BC:6

BC=4 BC=4

We know thatAB=DC=2BC AB=DC=2BC We replace the data we obtained (BC=4)

AB=DC=2×4=8 AB=DC=2\times4=8

As ABCD is a parallelogram, then all pairs of opposite sides are equal, therefore BC=AD=4

To find EC we use the formula:AABCD=AB×EC A_{ABCD}=AB\times EC

We replace the existing data:

24=8×EC 24=8\times EC

We divide the two sections by 824:8=8EC:8 24:8=8EC:8

3=EC 3=EC

Answer

3 cm