Find the perimeter and area of a quadrilateral ABCD in which BC=12cm ,CD=9cm,BD=15cm,DA= 17cm and angle ABD=90°.
Answers
Step-by-step explanation:
Consider Δ ABD
Using the Pythagoras theorem
AD
2
=AB
2
+BD
2
By substituting the values
17
2
=AB
2
+15
2
On further calculation
AB
2
=64
By taking out the square root
AB=
64
So we get
AB=8cm
We know that
Perimeter of quadrilateral ABCD=AB+BC+CD+AD
By substitution the values
Perimeter =8+12+9+17
By addition
Perimeter=46cm
We know that area of ΔABD=
2
1
×b×h
It can be written as:
Area of ΔABD=
2
1
×AB×BD
By substituting the values
Area of Δ ABC=
2
1
×8×15
On further calculation
Area of ΔABD=60cm
2
Consider ΔBCD
We know thatBC=12cm,CD=9cmandBD=15cm
It can be written as a=12cm,b=9cmandc=15cm
So we get
s=
2
a+b+c
s=
2
12+9+15
By division,
s=18cm
We know that
Area=
s(s−a)(s−b)(s−c)
By substituting the values
Area=
18(18−12)(18−9)(18−15
So we get
Area=
18×6×9×3
It can be written as
Area=
6×3×6×9×3
On further calculation
Area=6×3×3
By multiplication
Area=54cm
2
So the area of the shaded region=Area of ΔABD+ Area of ΔBCD
By substituting the values
Area of the quadrilateral ABCD=60+54=114cm
2
Therefore, the perimeter is 46cm and the area is 114cm
2
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