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.
Video Solution
Step-by-Step Solution
Let's call CE as X
According to the data
AB=x+2,AD=x−3
The perimeter of the parallelogram:
2(AB+AD)
38=2(x+2+x−3)
38=2(2x−1)
38=4x−2
38+2=4x
40=4x
x=10
Now it can be argued:
AD=10−3=7,CE=10
The area of the parallelogram:
CE×AD=10×7=70
Answer
70 cm²
Exercise #7
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
Video Solution
Step-by-Step Solution
The perimeter of the parallelogram is calculated as follows:
SABCD=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
We inut the data we know in the formula to calculate the perimeter:
PABCD=2BC+BC+2BC+BC
We replace the given perimeter in the formula and add up all the BC coefficients accordingly:
24=6BC
We divide the two sections by 6
24:6=6BC:6
BC=4
We know thatAB=DC=2BCWe replace the data we obtained (BC=4)
AB=DC=2×4=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
We replace the existing data:
24=8×EC
We divide the two sections by 824:8=8EC:8
3=EC
Answer
3 cm
Exercise #8
The parallelogram ABCD contains the rectangle AEFC inside it, which has a perimeter of 24.
AE = 8
BC = 5
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
16+2X=24
We isolate X:
2X=8
and divide by 2:
X=4
Now we can use the Pythagorean theorem to find EB.
(Pythagoras: A2+B2=C2)
EB2+42=52
EB2+16=25
We isolate the variable
EB2=9
We take the square root of the equation.
EB=3
The area of a parallelogram is the height multiplied by the side to which the height descends, that isAB×EC.
AB= AE+EB
AB=8+3=11
And therefore we will apply the area formula:
11×4=44
Answer
44
Exercise #9
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.
Video Solution
Step-by-Step Solution
To calculate the ratio between the sides we will use the existing figure:
AABCEAAED=31
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=2h×DE
We know that the area of the parallelogram is equal to:
AABCD=h×EC
We replace the data in the formula given by the ratio between the areas:
h×EC21h×DE=31
We solve by cross multiplying and obtain the formula:
h×EC=3(21h×DE)
We open the parentheses accordingly:
h×EC=1.5h×DE
We divide both sides by h:
EC=h1.5h×DE
We simplify to h:
EC=1.5DE
Therefore, the ratio between is: DEEC=1.51
Answer
1:1.5
Exercise #10
ABCD is a parallelogram.
Angle ACB is equal to angle EBC.
BF = 6
CE = 9
BF is perpendicular to DE.
Calculate the area of the parallelogram.
Video Solution
Step-by-Step Solution
Given that angle ACB is equal to angle CBE, it follows that AC is parallel to BE
since alternate angles between parallel lines are equal.
As we know that ABCD is a parallelogram, AB is parallel to DC and therefore AB is also parallel to CE since it is a line that continues DC.
Given that AC is parallel to BE and, in addition, AB is parallel to CE, it can be argued that ABCE is a parallelogram and, therefore, each pair of opposite sides in a parallelogram are parallel and equal.
From this it is concluded that AB=CE=9
Now we calculate the area of the parallelogram ABCD according to the data.
SABCD=AB×BF
We replace the data accordingly:
SABCD=9×6=54
Answer
54 cm²
Question 1
Below is a circle bounded by a parallelogram:
All meeting points are tangential to the circle. The circumference is 25.13.
All meeting points are tangential to the circle. The circumference is 25.13.
What is the area of the parallelogram?
Video Solution
Step-by-Step Solution
First, we add letters as reference points:
Let's observe points A and B.
We know that two tangent lines to a circle that start from the same point are parallel to each other.
Therefore:
AE=AF=3 BG=BF=6
And from here we can calculate:
AB=AF+FB=3+6=9
Now we need the height of the parallelogram.
We know that F is tangent to the circle, so the diameter that comes out of point F will also be the height of the parallelogram.
It is also known that the diameter is equal to two radii.
Since the circumference is 25.13.
Circumference formula:2πR We replace and solve:
2πR=25.13 πR=12.565 R≈4
The height of the parallelogram is equal to two radii, that is, 8.
And from here you can calculate with a parallelogram area formula:
AlturaXLado
9×8≈72
Answer
≈72
Exercise #12
The following is a circle enclosed in a parallelogram:
All meeting points are tangent to the circle. The circumference is 25.13.
What is the area of the zones marked in blue?
Video Solution
Step-by-Step Solution
First, we add letters as reference points:
Let's observe points A and B.
We know that two tangent lines to a circle that start from the same point are parallel to each other.
Therefore:
AE=AF=3 BG=BF=6
From here we can calculate:
AB=AF+FB=3+6=9
Now we need the height of the parallelogram.
We know that F is tangent to the circle, so the diameter that comes out of point F will also be the height of the parallelogram.
It is also known that the diameter is equal to two radii.
It is known that the circumference of the circle is 25.13.
Formula of the circumference:2πR We replace and solve:
2πR=25.13 πR=12.565 R≈4
The height of the parallelogram is equal to two radii, that is, 8.
And from here it is possible to calculate the area of the parallelogram:
Lado x Altura9×8≈72
Now, we calculate the area of the circle according to the formula:πR2
π42=50.26
Now, subtract the area of the circle from the surface of the trapezoid to get the answer:
72−56.24≈21.73
Answer
≈21.73
Exercise #13
ABCD is a parallelogram BFCE is a deltoid
What is the area of the parallelogram ABCD?
Video Solution
Step-by-Step Solution
First, we must remember the formula for the area of a parallelogram:Lado x Altura.
In this case, we will try to find the height CH and the side BC.
We start from the side
First, let's observe the small triangle EBG,
As it is a right triangle, we can use the Pythagorean theorem (
A2+B2=C2)
BG2+42=52
BG2+16=25
BG2=9
BG=3
Now, let's start looking for GC.
First, remember that the deltoid has two pairs of equal adjacent sides, therefore:FC=EC=9
Now we can also do Pythagoras in the triangle GCE.
GC2+42=92
GC2+16=81
GC2=65
GC=65
Now we can calculate the side BC:
BC=BG+GT=3+65≈11
Now, let's observe the triangle BGE and DHC
Angle BGE = 90° Angle CHD = 90° Angle CDH=EBG because these are opposite parallel angles.
Therefore, there is a ratio of similarity between the two triangles, so:
BGHD=GEHC
BGHD=37.5=2.5
EGHC=4HC=2.5
HC=10
Now that there is a height and a side, all that remains is to calculate.
10×11≈110
Answer
≈110
Exercise #14
The parallelogram ABCD is shown below.
BC is the diameter of the circle whose circumference is equal to 10π cm.
ECFD is a rhombus whose area is 24 cm².
What is the area of ABCD?
Video Solution
Step-by-Step Solution
Let's try to calculate the area in two ways.
In the first method, we will try to use the rhombus ECFD:
Let's try to calculate according to the formula area=DC×hDC
We will lower a height to DC and see that we do not have enough data to calculate, so we will not be able to calculate the area of the parallelogram using the rhombus.
In the second method , we will try to use the circle:
area=BC×hBCWe will lower a height to BC and see that we do not have enough data to calculate, so we will not be able to calculate the area of the parallelogram using the circle.
From this it follows that we do not have enough data to calculate the area of parallelogram ABCD and therefore the exercise cannot be solved.
Answer
It is not possible to calculate.
Exercise #15
Calculate the area of the parallelogram based on the data in the figure: