Question

2. Find the perimeter and area of a quadrilateral
ABCD in which BC = 12 cm, CD = 9 cm, BD = 15 cm,
DA = 17 cm and ABD = 90°.​

Answer 1

Answer:

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 that BC=12cm,CD=9cm and BD=15cm

It can be written as a=12cm, b=9cm and c=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