# quadrilateral area and diagonals

7. Figure 4 shows a survey of a building which forms a quadrilateral ABCD

Calculate a) the length of the diagonals - AC and BD

b) the area of the plot ABCD

Please see attachment for diagrams.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

7.a

AC^2 = AB^2 + BC^2 - 2*AB*BC*cos(65)

=> AC = sqrt(21.4^2 + 29.6^2 - 2*21.4*29.6*cos(65)) = 28.26 --ANSWER

To estimate BD:

AC^2 = AD^2 + DC^2 - 2 * AD * DC * cos(ADC) = AB^2 + BC^2 - 2*AB*BC*cos(65)

=> 2*AD*DC*cos(ADC) = AD^2 + DC^2 - AB^2 - BC^2 + 2*AB*BC*cos(65)

=> cos(ADC) = (AD^2 + DC^2 - AB^2 - BC^2 + 2*AB*BC*cos(65))/(2*AD*DC)

=> cos(ADC) = (19.3^2 + 17.9^2 - 21.4^2 - 29.6^2 + 2*21.4*29.6*cos(65))/(2*19.3*17.9) = - 0.153

=> angle ADC = cos-1(-0.153) = 98.8 degree == 99 degree

Hence,

angle DAB + angle DCB = 360 - 65 - 99 = 196 degree

From triangle ABC:

AC/sin(ABC) = AB/sin(ACB)

=> sin(ACB) = AB*sin(ABC)/AC = 21.4 *sin(65)/28.26 = 0.686

=> angle ACB == 43.5 degree

from triangle ADC:

AC/sin(ADC) = AD/sin(ACD)

=> sin(ACD) = AD*sin(ADC)/AC = 19.3*sin(99)/28.26 = 0.675

=> angle ACD = 42.5 degree

Therefore, angle BCD = angle ACB + angle ACD = 43.5+42.5 = 86 degree

Triangle BCD:

BD = sqrt(BC^2 + CD^2 - 2*BC*CD*cos(BCD))

=> BD = sqrt(29.6^2 + 17.9^2 - 2*29.6*17.9*cos(86)) = 33.5 --ANSWER.

https://brainmass.com/math/geometric-shapes/quadrilateral-area-diagonals-540023