The answer is 5.1 units. This is easier than it looks. And I am glad that I have reviewed how to apply AA similarity for deducing the area which is a form of subtraction. I shall use that for practice!!!
The white triangle base = 2 cm the upper right triangle and lower right triangle are similar with a linear ratio of 3 : 2 the side of the square = x x^2 + (1 - 2x/3)^2 = 3^2 x^2 + x^2/9 = 3^2 10x^2 = 9(3^2) = 81 x^2 = 81/10 shaded area = (81/10) cm^2 - 3 cm^2 = (51/10) cm^2
1/EF=2 2/Angle BAF=angle EFC (having sides perpendicular) --> the two right triangles BAF and CFE are similar and the relevant ratio is 3/2 ( the ratio of the relevant hypotenuses) --> CF/a= 2/3--> CF=2a/3-> AF = a/3 Using Pythagorean theorem sqa+sq a/9 = 9 --> sqa=8.1 Area of the purple region = 5.1 sq cm😅😅o
Let x is the side of the square . Area of the AEF=3 cm^2 1/2(AF)(EF)=3 1/2(3)(EF)=3 So EF=2 cm ∆ABF~∆FCE So x/3=CF/EF BF=√9-x^2 CF=x-√9-x^2 x/3=(x-√9-x^2)/2 2x=3x-3√9-x^2 3√9-x^2=x 9(9-x^2)=x^2 81-9x^2=x^2 10x^2=81 So x^2=81/10 cm^2=8.1 cm^2 is area of the square. So Purple shaded area=8.1-3=5.1 cm^2 ❤❤❤
Triangle ∆EFA: A = bh/2 = FA(EF)/2 3 = 3(EF)/2 EF/2 = 1 EF = 2 Let s be the side length of square ABCD. As ∠EFA = 90°, then ∠AFB and ∠CFE are complementary angles and sum to 90°. As ∆ECF and ∆FBA are right triangles, then each contains both of the complementary angles and thus ∆ECF and ∆FBA are similar. CF/BA = FE/AF CF/s = 2/3 CF = 2s/3 CB = CF + FB s = 2s/3 + FB FB = s/3 Triangle ∆FBA: FB² + BA² = AF² (s/3)² + s² = 3² s²/9 + s² = 9 10s²/9 = 9 10s² = 81 s² = 81/10 Shaded area: A = s² - 3 A = 81/10 - 3 = 51/10 [ A = 5.1 cm² ]
Solution: A ∆AEF = ½ b h 3 = ½ b 3 3b/2 = 3 b = 3 . 2/3 b = 2 cm Considering "x" as the side of the square and knowing that triangles ABF and CEF are similar ("α" and "β" are complementary angles), we have proportions: AB = x AF = 3 EF = 2 CF = y x/3 = y/2 y = 2x/3 BF = x - y BF = x - 2x/3 BF = x/3 Applying Pythagorean Theorem in ∆ ABF, we have: x² + (x/3)² = 3² x² + x²/9 = 9 (×9) 9x² + x² = 81 10x² = 81 x² = 8,1 cm² Purple Area = Square ABCD Area - Triangle AEF Area Purple Area = 8,1 - 3 Purple Area = 5,1 cm² ✅
Let's find the area: . .. ... .... ..... From the given values for the right triangle AEF we can conclude: A(AEF) = (1/2)*AF*h(AF) = (1/2)*AF*EF ⇒ EF = 2*A(AEF)/AF = 2*(3cm²)/(3cm) = 2cm The purple triangles ABF and CEF are similar right triangles: ∠ABF = ∠ECF = 90° ∧ ∠BAF = ∠CFE ⇒ ∠AFB = ∠CEF Therefore we can conclude: CF/AB = EF/AF = (2cm)/(3cm) = 2/3 ⇒ CF = 2*AB/3 Now we apply the Pythagorean theorem to the right triangle ABF. With s being the side length of the square ABCD we obtain: AF² = AB² + BF² AF² = AB² + (BC − CF)² AF² = AB² + (BC − 2*AB/3)² AF² = s² + (s − 2*s/3)² = s² + (s/3)² = s² + s²/9 = 10*s²/9 ⇒ A(ABCD) = s² = 9*AF²/10 = 9*(3cm)²/10 = 8.1cm² Now we are able to calculate the area of the purple region: A(purple) = A(ABCD) − A(AEF) = 8.1cm² − 3cm² = 5.1cm² Best regards from Germany
My method differed slightly. I wrote some unnecessary info which I deleted before posting. AEF has sides 2, 3, sqrt(13). However, using AFE's right angle as a guide, it appears that ABF and CFE are similar. FCE's side lengths are 2/3 of ABF's due to EF and FA corresponding sides being 2 and 3 respectively. FC = (2/3)x. Area ABF = (x*(1/3)x)/2 = (1/6)x^2 Due to the 2/3 fraction, ECF is 4/9 of that so (4/54)x^2 = (2/27)x^2 ABF + ECF = (1/6)x^2 + (2/27)x^2 = 9/54 + 4/54 = (13/54)x^2 EC is (1/3)*(2/3) = 2/9 Now ADE: (x*(7/9)x)/2 = (7/18)x^2 The three purple triangles are (13/54)x^2 + (7/18)x^2 = ((13 + 21)/54)x^2 = (34/54)x^2 = (17/27)x^2 This means that the remaining(10/27)x^2 = 3 If (10/27)x^2 = 3, then (17/27)x^2 = 3*(17/10) = 51/10 = 5.1cm^2
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You are very welcome!
Thanks for the feedback ❤️
The answer is 5.1 units. This is easier than it looks. And I am glad that I have reviewed how to apply AA similarity for deducing the area which is a form of subtraction. I shall use that for practice!!!
The white triangle base = 2 cm
the upper right triangle and lower right triangle are similar with a linear ratio of 3 : 2
the side of the square = x
x^2 + (1 - 2x/3)^2 = 3^2
x^2 + x^2/9 = 3^2
10x^2 = 9(3^2) = 81
x^2 = 81/10
shaded area = (81/10) cm^2 - 3 cm^2 = (51/10) cm^2
(3)^2 (3)^2={9+9}=18 360°ABCD/18=2ABCD (ABCD ➖ 2ABCD+1).
1/EF=2
2/Angle BAF=angle EFC (having sides perpendicular) --> the two right triangles BAF and CFE are similar and the relevant ratio is 3/2 ( the ratio of the relevant hypotenuses)
--> CF/a= 2/3--> CF=2a/3-> AF = a/3
Using Pythagorean theorem
sqa+sq a/9 = 9 --> sqa=8.1
Area of the purple region = 5.1 sq cm😅😅o
EF=2*3/3=2--->. Si AB=c--->FC=2c/3---> BF=c/3---> c²+(c/3)²=3²---> c²=81/10---> Área sombreada =(81/10)-3=51/10.
Gracias y saludos
Thank you!
You are very welcome!
Thanks for the feedback ❤️
Let x is the side of the square .
Area of the AEF=3 cm^2
1/2(AF)(EF)=3
1/2(3)(EF)=3
So EF=2 cm
∆ABF~∆FCE
So x/3=CF/EF
BF=√9-x^2
CF=x-√9-x^2
x/3=(x-√9-x^2)/2
2x=3x-3√9-x^2
3√9-x^2=x
9(9-x^2)=x^2
81-9x^2=x^2
10x^2=81
So x^2=81/10 cm^2=8.1 cm^2 is area of the square.
So Purple shaded area=8.1-3=5.1 cm^2
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❤
Triangle ∆EFA:
A = bh/2 = FA(EF)/2
3 = 3(EF)/2
EF/2 = 1
EF = 2
Let s be the side length of square ABCD. As ∠EFA = 90°, then ∠AFB and ∠CFE are complementary angles and sum to 90°. As ∆ECF and ∆FBA are right triangles, then each contains both of the complementary angles and thus ∆ECF and ∆FBA are similar.
CF/BA = FE/AF
CF/s = 2/3
CF = 2s/3
CB = CF + FB
s = 2s/3 + FB
FB = s/3
Triangle ∆FBA:
FB² + BA² = AF²
(s/3)² + s² = 3²
s²/9 + s² = 9
10s²/9 = 9
10s² = 81
s² = 81/10
Shaded area:
A = s² - 3
A = 81/10 - 3 = 51/10
[ A = 5.1 cm² ]
Solution:
A ∆AEF = ½ b h
3 = ½ b 3
3b/2 = 3
b = 3 . 2/3
b = 2 cm
Considering "x" as the side of the square and knowing that triangles ABF and CEF are similar ("α" and "β" are complementary angles), we have proportions:
AB = x
AF = 3
EF = 2
CF = y
x/3 = y/2
y = 2x/3
BF = x - y
BF = x - 2x/3
BF = x/3
Applying Pythagorean Theorem in ∆ ABF, we have:
x² + (x/3)² = 3²
x² + x²/9 = 9 (×9)
9x² + x² = 81
10x² = 81
x² = 8,1 cm²
Purple Area = Square ABCD Area - Triangle AEF Area
Purple Area = 8,1 - 3
Purple Area = 5,1 cm² ✅
Let's find the area:
.
..
...
....
.....
From the given values for the right triangle AEF we can conclude:
A(AEF) = (1/2)*AF*h(AF) = (1/2)*AF*EF ⇒ EF = 2*A(AEF)/AF = 2*(3cm²)/(3cm) = 2cm
The purple triangles ABF and CEF are similar right triangles:
∠ABF = ∠ECF = 90° ∧ ∠BAF = ∠CFE ⇒ ∠AFB = ∠CEF
Therefore we can conclude:
CF/AB = EF/AF = (2cm)/(3cm) = 2/3 ⇒ CF = 2*AB/3
Now we apply the Pythagorean theorem to the right triangle ABF. With s being the side length of the square ABCD we obtain:
AF² = AB² + BF²
AF² = AB² + (BC − CF)²
AF² = AB² + (BC − 2*AB/3)²
AF² = s² + (s − 2*s/3)² = s² + (s/3)² = s² + s²/9 = 10*s²/9
⇒ A(ABCD) = s² = 9*AF²/10 = 9*(3cm)²/10 = 8.1cm²
Now we are able to calculate the area of the purple region:
A(purple) = A(ABCD) − A(AEF) = 8.1cm² − 3cm² = 5.1cm²
Best regards from Germany
*Solução:*
A[APF] = 3×EF/2 = 3 →EF=2.
Seja ∠BAF=α. Assim ,
∠BFA=90° - α → ∠EFC=α. Daí,
AB = 3 cos α, BF = 3 sen α e CE= 2 cos α.
Como ABCD é um quadrado, então
AB = BF + FC
3 cos α = 3 sen α + 2 cos α
cos α = 3 sen α. Ora,
sen² α + cos² α = 1
(cos α/3)² + cos² α = 1
cos² α/9 + cos² α = 1
*_cos² α = 9/10._*
A [ABCD]=AB²= (3 cos α)²=
9 cos² α = 81/10.
Portanto,
A[painted] = 81/10 - 3 = (81 - 30)/10
*A[painted] = 51/10 = 5,1 cm²*
3*FE/2=3 FE=2
ABF∞FCE EC=2x FC=2y FB=3x AB=3y
3y=2y+3x y=3x AB=9x
(3x)²+(9x)²=3² 90x²=9 x²=1/10
Purple shaded area : 9x*9x - 3 = 81x² - 3 = 81/10 - 3 = 51/10 = 5.1cm²
Thanks sir ❤❤
You're very welcome! ❤️
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Amor e orações dos EUA! 😀
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My method differed slightly. I wrote some unnecessary info which I deleted before posting.
AEF has sides 2, 3, sqrt(13).
However, using AFE's right angle as a guide, it appears that ABF and CFE are similar.
FCE's side lengths are 2/3 of ABF's due to EF and FA corresponding sides being 2 and 3 respectively.
FC = (2/3)x.
Area ABF = (x*(1/3)x)/2 = (1/6)x^2
Due to the 2/3 fraction, ECF is 4/9 of that so (4/54)x^2 = (2/27)x^2
ABF + ECF = (1/6)x^2 + (2/27)x^2 = 9/54 + 4/54 = (13/54)x^2
EC is (1/3)*(2/3) = 2/9
Now ADE: (x*(7/9)x)/2 = (7/18)x^2
The three purple triangles are (13/54)x^2 + (7/18)x^2 = ((13 + 21)/54)x^2
= (34/54)x^2 = (17/27)x^2
This means that the remaining(10/27)x^2 = 3
If (10/27)x^2 = 3, then (17/27)x^2 = 3*(17/10) = 51/10 = 5.1cm^2
3×3=9-1=8squrooth × squrooth=8
5.1
Metric- is for people that count on their fingers…
very difficult. NO idea at all.😂😂😂😂😂😂😂