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    In quadrilateral ABCD, AB = 17 cm, BC = 8 cm, CD = 9 cm, AD = 12 cm, and AC = 15 cm. What is the area (in cm²) of the quadrilateral?​
    Question

    In quadrilateral ABCD, AB = 17 cm, BC = 8 cm, CD = 9 cm, AD = 12 cm, and AC = 15 cm. What is the area (in cm²) of the quadrilateral?​

    A.

    105

    B.

    114

    C.

    118

    D.

    121

    Correct option is B

    Given:

    AB = 17 cm, BC = 8 cm, CD = 9 cm,

    AD = 12 cm, AC = 15 cm

    Formula Used:

    Heron's formula for the area of a triangle with sides a, b, and c is:

    Area = s(sa)(sb)(sc)\sqrt{s(s-a)(s-b)(s-c)}

    where s is the semi-perimeter, given by:

    s = a+b+c2\frac{a + b + c}{2}​​

    Solution: 

    Area of Triangle ABC

    Sides: AB = 17, BC = 8, and AC = 15

    Semi-perimeter:

    s =17+8+152 \frac{17 + 8 + 15}{2}​ = 20 cm

    Area of ABC = 20(2017)(208)(2015)\sqrt{20(20 - 17)(20 - 8)(20 - 15)}​​

    =20×3×12×5= \sqrt{20 \times 3 \times 12 \times 5}​ 

    =3600=60 cm2= \sqrt{3600} = 60 \text{ cm}^2​​

    Area of Triangle ACD

    Semi-perimeter  = 12+9+152=18 cm\frac{12 + 9 + 15}{2} = 18 \text{ cm}​​

    Area of ACD = 18(1812)(189)(1815)\sqrt{18(18 - 12)(18 - 9)(18 - 15)}​​

    =18×6×9×3= \sqrt{18 \times 6 \times 9 \times 3}​​

    =2916=54 cm2= \sqrt{2916} = 54 \text{ cm}^2​​

    Total Area  of quadrilateral ABCD = 60 + 54 = 114 cm2 

    Alternate Solution: 

    In triangle ABC 

    Sides: AB = 17, BC = 8, and AC = 15 

    Form triplet , 

    So, Area of triangle ABC = 12×8×15=60 cm2\frac{1}2\times 8 \times 15 = 60 \ cm^2 

    ​Area of Triangle ACD:

    Sides: AC = 15,DC = 9, and AD = 12 (triplet)

    Area of ACD = 12×9×12=54 cm2\frac12\times 9\times 12 = 54 \ cm^2​  

    Area of ABCD = 60 + 54 = 114 cm2

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