The area is special

AB=3,BC=4AB=3,BC=4. AE=AB2AE=\frac{AB}{2} and BD=BC2BD=\frac{BC}{2}. ABCABC is a right triangle. Can you tell me the area of BDFEBDFE?

#Geometry

Note by Fahim Muhtamim
1 year, 7 months ago

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1 vote

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Comments

If it is a right triangle, the area of BDFE BDFE is 3 3 , which is half the area of the triangle.

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import math

# Coordinates of points

thetaABC = (math.pi/2.0)  # angle ABC

Bx = 0.0
By = 0.0

Cx = 4.0
Cy = 0.0

Ax = 3.0*math.cos(thetaABC)
Ay = 3.0*math.sin(thetaABC)

Ex = Ax/2.0
Ey = Ay/2.0

Dx = Cx/2.0
Dy = Cy/2.0

########################################

# Use line intersection equations to find location of point F

b1 = Ey    # y intercept
b2 = Ay

m1 = -Ey/Cx
m2 = -Ay/Cx  # slope

# m1*Fx + b1 = m2*Fx + b2
# Fx*(m1-m2) = b2 - b1

Fx = (b2-b1)/(m1-m2)
Fy = m1*Fx + b1


########################################

# Vectors from B to E,F,D

v1x = Ex - Bx
v1y = Ey - By

v2x = Fx - Bx
v2y = Fy - By

v3x = Dx - Bx
v3y = Dy - By

########################################

# Use cross products to find areas.....
#  ......of triangles EBF and DBF

cross1 = v1x*v2y - v1y*v2x
cross2 = v2x*v3y - v2y*v3x

Area = 0.5*math.fabs(cross1) + 0.5*math.fabs(cross2)

print Area
# Area = 3

Steven Chase - 1 year, 7 months ago

But its a geometry note not computer science.

Fahim Muhtamim - 1 year, 7 months ago

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Is angle ABC allowed to take any value, or is it supposed to be a right triangle?

Steven Chase - 1 year, 7 months ago

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Its a right angle.

Fahim Muhtamim - 1 year, 7 months ago

By GeoGebra,I've got this 2

Fahim Muhtamim - 1 year, 7 months ago

Let, FD=aFD=a, ED=bED=b and DE=cDE=c

By Pithagorus' theorem, we get

AD=22+32=13AD=\sqrt{2^2+3^2}=\sqrt{13}

So, a=133a=\frac{\sqrt{13}}{3}

By the same way, b=736b=\frac{\sqrt{73}}{6}.

And c=322+22=52c=\sqrt{{\frac{3}{2}}^2+2^2}=\frac{5}{2}

Then let,s=a+b+c2s=\frac{a+b+c}{2}

And the area of triangle DEF=s(s−a)(s−b)(s−c)=12DEF=\sqrt{s(s-a)(s-b)(s-c)}=\frac{1}{2}

And thr area of BED=32BED=\frac{3}{2}

And the total area is 12+32=2\frac{1}{2}+\frac{3}{2}=2

Fahim Muhtamim - 1 year, 7 months ago
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