A park with flower plants is to be developed within a quadrilateral with points A(0, −1), B(6, 7), C(−2, 3) and D(8, 3) as vertices and AB and CD as diagonals. Show that AB and CD bisect each other and AD2 + DB2 = AB2. Find the area of the park. (All distances are in km)
Answers
Answer:
Proved
40 km²
Step-by-step explanation:
A park with flower plants is to be developed within a quadrilateral with points A(0, −1), B(6, 7), C(−2, 3) and D(8, 3) as vertices and AB and CD as diagonals. Show that AB and CD bisect each other and AD2 + DB2 = AB2. Find the area of the park. (All distances are in km)
AB A (0,-1) , B (6,7)
y = mx + c
m = (7-(-1))/(6-0) = 8/6 = 4/3
y = 4x/3 + c
3y = 4x + 3c
3(-1) = 4*0 + 3c => 3c = -3
3y = 4x - 3 - Eq 1 ( AB)
CD (-2,3) (8,3)
m = (3-3)/(8-(-2) = 0
y = c
y = 3 = Eq 2 (CD)
putting y = 3 in eq 1
3*3 = 4x - 3
=> 4x = 12
=> x = 3
Point of intersection of AB & CD
= (3,3)
if AB & CD bisect at this point then
coordinates of AB bisector
((6+0)/2 , (7+(-1))/2) = (3 , 3) Same as found earlier
coordinates of CD bisector
((-2+8)/2 , (3+3)/2) = (3 , 3) Same as found earlier
Hence proved that AB and CD bisect each other
AD² = (8-0)² + (3-(-1))² = 64 + 16 = 80
DB² = (6-8)² + (3-7)² = 4 + 16 = 20
AB² = (6-0)² + (7-(-1))² = 36 + 64 = 100
AD² + DB² = 80 + 20 = 100 = AB²
Hence Proved that
AD² + DB² = AB²
ΔADB is right angled triangle at point D and Diagonal bisect each other so its an rectangular filed
with Length & Width = AD & DB = √80 & √20
Area of ABCD = √80 * √20 = √1600 = 40 km²