Can you find area of the Pink Shaded Trapezoid? | Trapezoid | (Trapezium) |
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- Опубликовано: 9 май 2024
- Learn how to find the area of the Pink Trapezoid. Important Geometry and algebra skills are also explained: Trapezoid; Trapezium; Trapezoid area formula; Trigonometry; Square. Step-by-step tutorial by PreMath.com
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First comment and first like, you can pin it?
Done!
Thank you sir
Tan 30 = a /(100 - a ).
a = tan 30 x (100 - a ).
a = 100 tan 30 - a tan 30.
a + a tan 30 = 100 tan 30.
a (1 + tan 30 ) =100 tan 30.
a = 100 tan 30 /1 + tan 30.
a = 36.60254.
Area = 1/2 ( 36.60254 +100 ) x 36.60254.
2500.
Excellent!
Thanks for sharing ❤️
You did it exatly like me !
1. Connect A&C. Diagonal AC is the hypotenuse of the isosceles triangle ABC. Angle CAB=45 deg., angle ACB=180-45-30=105 deg. From sine teorem for triangle ABC: 100/sin 105 deg=AC/sin 30 deg.
Connect C to E
AD=DC=CE=EA=x
In ∆BEC
BE=100-x
Tan(30°)=x/100-x
x=50√3-50
So area trapezoid=1/2(100+50√3-50)(50√3-50)=2500 square units.❤❤❤ Thanks sir.
Excellent!
You are very welcome!
Thanks for sharing ❤️
bc^2 = a^2 + (100-a)^2 -> (2a)^2 = a^2 + 100^2 -200a + a^2 -> 4a^2 - 2a^2 +200a = 10000 -> 2a^2 + 200a = 10000 -> a^2 + 100a = 5000
abcd = 1/2(100+a)*a = 1/2( a^2 + 100a) -> 1/2(5000) = 2500
Excellent!
Thanks for sharing ❤️
Nice problem, nice solving...
a/(100-a)=tg30=1/√3...100/a=√3+1..a=100/(√3+1)..Apink=(100+a)a/2=
Excellent!
Thanks for sharing ❤️
Solution is okay. Thanks!
You are very welcome!
Thanks for the feedback ❤️
Can you do more of this sir they are helpful
Drop a perpendicular from C to E on AB. As CD = DA and all internal angles are 90°, AECD is a square. Let CD = DA = AE = CE = x.
As ∠CEB = 90°, ∆CEB is a right triangle, and as ∠EBC = 30°, that makes it a 30-60-90 special right triangle, where ∠BCE = 60°, BC = 2CE and EB = √3CE.
As CE = AE, EB = √3AE, thus:
AB = AE + EB
100 = AE + √3AE
100 = x + √3x = x(1+√3)
x = 100/(1+√3)
x = 100(1-√3)/(1+√3)(1-√3)
x = 100(1-√3)/(1-3)
x = 100(√3-1)/2 = 50(√3-1)
Pink Trapezoid ABCD:
A = h(a+b)/2 = x(x+100)/2
A = ((50(√3-1))² + 100(50(√3-1)))/2
A = (2500(3-2√3+1) + 5000(√3-1))/2
A = 1250(4-2√3) + 2500(√3-1)
A = 5000 - 2500√3 + 2500√3 - 2500
A = 2500 sq units
Excellent!
Thanks for sharing ❤️
Yesssss!!!!! I got it! Cheers!
tan(30)=x/(100-x)=0,5773
x=36,602
Area=x2/2+50x
After substitution and calculation Area=2500s.u
Excellent!
Thanks for sharing ❤️
Draw a segment thru C and a point E on segment AB, such that segment CE is perpendicular to the bases of trapezoid ABCD.
This forms a square ADCE because AD = CD. Label the side length of square ADCE as s.
Segment CE also forms △BEC, a special 30°-60°-90° right triangle.
BE = (CE)√3
= s√3
So, AB = AE + BE = s + s√3 = 100.
s + s√3 = 100
s(√3 + 1) = 100
s = 100/(√3 + 1)
= 100/(√3 + 1) * (√3 - 1)/(√3 - 1)
= (100√3 - 100)/2
= 50√3 - 50
A = h[(a + b)/2]
= (50√3 - 50)[(50√3 - 50 + 100)/2]
= (50√3 - 50) * (50√3 + 50) * 1/2
= (7500 - 2500) * 1/2
= 1/2 * 5000
= 2500
So, the area of the pink trapezoid is 2,500 square units.
Excellent!
Thanks for sharing ❤️
Before watching: I arrive at 2.500 square units. Approach: the left part is a square, the right part is a 30-60-90 triangle. Side relations: 1-sqrt(3)-2. to arrive at the side of the square, divide 100 by (1+sqrt(3)). The side of the square is approx 36,6. From here calculate the areas of the square and the triangle, which totals to 2500.
After watching: exactly what I did, though I didn‘t have pen and paper, so I couldn‘t do the pretty steps with rationalizing the denominator etc, but simply used the calculator and my iPhone…)
Mr Philip excellent work
Thanks. To make more clear sol, we may draw EF so that F is on BC and angle BEF is 30 degrees.
Then angle CEF will be 60 degrees.
🔺 CEF will be an equilateral 🔺 .This means
CE =CF = EF
🔺 BEF is an isosceles 🔺 and EF=BF
Then CE=CF=EF=BF
If CE=a
Then BC =CF+EF=CE+CE=2a
Thanks for sharing ❤️
Good puzzle. Fun to do.
Excellent '
Many thanks dear!❤️
Great video again Sire ! I used tan (30) = x / ( 100 - x ) when x = CE and EB = 100 - x . I got for x nearly 36,6 and for the pink area
A = 2499,78 square units. But using the third binomal formula was more elegant than my solution. You got exactly 2500
sqare units.
Thank you
I used trig. Tan and cosine formula to calculate the length of the diameter. It is a shorter and easier altertive. I believe
Amazing !!
Thank you! Cheers!🌹❤️
Excellent! 🙂
Thank you! Cheers!❤️
100=(sqrt(3)+1)a, a=100/(sqrt(3)+1)=50(sqrt(3)-1), therefore the area is (1/2)×(100+50(sqrt(3)-1))×50(sqrt(3)-1)=(2500/2)(sqrt(3)+1)(sqrt(3)-1)=2500.🎉
Excellent!
Thanks for sharing ❤️
I missed a few tricks there, including the initial factorisation. However, even using decimal approximations and a calculator I got to 2499.9ish.
Let's give this one a try:
100 = s(square) + x
x = s√3 (since there is a 30-60-90° triangle)
100 = s√3 + s
s(√3 + 1) = 100
s = 100/(√3 + 1) = 100 (√3 - 1) / ((√3 + 1)(√3 - 1)) = 100 (√3 - 1) / (3 - 1) = 100 (√3 - 1) / 2 = 50 (√3 - 1)
A = A(square) + A(triangle) = s² + 1/2 * s√3 * s = s² + 1/2 * s² * √3 = s² (1 + 1/2 √3)
A = (50 (√3 - 1))² (1 + 1/2 √3) = (2500 (3 - 2√3 + 1)) (1/2 (2 + √3)) = (2500 (4 - 2√3)) (1/2 (2 + √3)) = 1250 (4 - 2√3) (2 + √3) = 1250 (8 + 4√3 - 4√3 - 6) = 1250 * 2 = 2500 square units
Excellent!
Thanks for sharing ❤️
so great
Glad to hear that!
Thanks for the feedback ❤️
Thanks Sir
That’s wonderful method .
I am answer if it is possible find the area by plusing the area of triangle and square area.
With glades
Much simple.
S=2500
Excellent!
Thanks for sharing ❤️
Thanks I solved it too 😊😊😊❤❤❤
It was tasty 😋
I got the right answer with a different way:
I M P O R T A N T :
• my calculations are carried out with 6 digits after the decimal point because I use a trigonometric function (tangent)
• the point "E" is on the line AB, at the opposite of C
get x = CE
tan(30) = x/(100 - x)
tan(30)·(100 - x) = x
0.577350·(100 - x) = x
57.7350 - 0.577350x = x
x + 0.577350x = 57.7350
1.577350x = 57.7350
x = 57.7350/1.577350
x = 36.602529
trapezoid area:
area = x² + (x·(100 - x))/2
area = 36.602529² + (36.602529·(100 - 36.602529))/2
-----------------------------
| area = 2499.99 |
-----------------------------
🙂
Maybe it's a cultural difference, but I was taught never to refer to the hypotenuse of a right triangle as a "leg."
Good solution though.
Thanks for the feedback ❤️
Kind regards🌹
x+x√3=100.
X=100/(1+√3)
X=50(-1+√3)
2❤️=(X+100)X
❤️=50²
❤️=2500
💙💚💛💜❤️🖤.
Hamas=Résistance ❤️🖤
Thanks for sharing ❤️
Anxhela besoj se e ke me te lehte tani?
Easy once you see the square and the 30-60-90 triangle
Thanks for the feedback ❤️
A little bit late but here I am!!
1) tan(30º) = sqrt(3) / 3
2) AD = CD = X
3) So, CC' = X
4) BC' = (100 - X)
5) CC' / BC' = tan(30º)
6) X / (100 - X) = tan(30º)
7) X / (100 - X) = sqrt(3) / 3
8) X = 50 * (sqrt(3) - 1) ~ 36,6 lin un
9) T = B + b * (h/2)
10) T = (100 + 36,6) * 36,6 / 2
12) T = 136,6 * 18,3
13) T = 2.499,78 sq un
14) My Best Answer is : The Pink Trapezoid Area is approx. equal to 2.500 Square Units.
Would 2499.78 square units be acceptable? That is what I got using fewer steps.
Yes! Close enough.
Thanks for sharing ❤️
2500
Excellent!
Thanks for sharing ❤️
Tan 30 = a/(100-a). Go from there. Easy stuff.
Thanks for the feedback ❤️
I think without using trapezium area formula , area of trapezium can be found in this problem
Thanks for the feedback ❤️
Shoh vetem figuren. Jepen te 4 kendet e nje trapezi qe jane 90, 30, 150 dhe 90 grade. Jepet baza e madhe 100. Zbato formuat dhe gjindet sip. e trapezit. Po si mbajte mend, i nxjerr duke formuar trek. Dhe nje katror me brinje ma bazen e vogel te trapezir. Diferenca ndermjet tyre edhte kateti i trk. kendrejte me kende 30 dhe 60 grade. Tani vetem .....
Let's find the area:
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First of all we add point E on AB such that AE=CD. Then ADCE is a square and since ∠AEC=∠BEC=90°, the triangle BCE is a 30°-60°-90° triangle. Therefore we can conclude immediately:
BE = √3*CE
So from the known length AB we obtain:
AB = AE + BE = CE + BE = CE + √3*CE = CE*(1 + √3)
⇒ CE = AB/(1 + √3) = (1 − √3)*AB/[(1 + √3)(1 − √3)] = (√3 − 1)*AB/2
Now we are able to calculate the area of the trapezoid:
A(ABCD)
= (1/2)*(AB + CD)*AD
= (1/2)*(AB + CE)*CE
= (1/2)*[AB + (√3 − 1)*AB/2]*(√3 − 1)*AB/2
= [1 + (√3 − 1)/2]*(√3 − 1)*AB²/4
= [(√3 + 1)/2]*(√3 − 1)*AB²/4
= AB²/4
= 100²/4
= 2500
Best regards from Germany
Excellent!
Thanks for sharing ❤️
Ur explanation confusing bad, u let it looks very difficult, it’s too long 🤦♀️