Question

Two points A and C have coordinates (1,3) and (7,7).
a. Find the equation of the perpendicular bisector of AC
b. B is a point on the y-axis which is equidistant from A and C and ABCD is a rhombus
find point B and Point D
c. find the area of the rhombus ABCD
d. calculate the perpendicular distance of A from BC

Answer 1

i have found point B (0,11) but need point D!

Answer 2

DIstance of B to the perpendicular of AC = Distance of D from AC . And the lines AC and BD cuts each other perpendicularly . Area of rhombus is 0.5( AC * BD)

Answer 3

how do you find point D then?

Answer 4

Well you have to find the area of the rhombus and you need to to mulitply the two diagonal length together with half , and you can find the distance of B to the diagonal using pythagoras , the length of BD is twice of that .

Answer 5

i cant find the area if i dont have all my points can I?

Answer 6

a) here slope of AC is (7-3)/(7-1) =2/3
midpoint of AC is (1+7)/2,(7+3)/2=>(4,5)
equation of line perpendicular bisector to ac is (y-5)= -3/2 (x-4)
or 2y-10=-3x+12
or 3x+2y=22

Answer 7

b) bcoz B is a pt equidistant from A and C hence B must lie on the perpendicular bisector of AC i.e B lies on 3x+2y=22
let pt B be (0,b)[as B lies on y-axis
hence we have
3.0+2.b=22
or b=11
hence cordinate of B is (0,11)

Answer 8

and what about the coordinate of D?

Answer 9

let the cordinates of D be (p,q) hence midpoint of BD and AC must coincide as ABCD is a rhombus
thus midpoint of BD must be (4,5)
i.e (0+p)/2,(11+q)/2 is (4,5
or (p/2)=4 and (11+q)/2=5
solving we have p=8 and q=-1
hence D is (8,-1)

Answer 10

AC=sqrt((7-1)^2 +(7-3)^2)=sqrt(52)=2*sqrt(13)
BD=sqrt((8-0)^2+(11+1)^2)=sqrt(208)=4*sqrt(13)
hence area of ABCD =1/2 *AC*BD =1/2 *2*sqrt(13)*4*sqrt(13) = 52sq units

Answer 11

now eq of line joining B(0,11) and C(7,7)
is y-11=-4/7 *x
or 4x+7y=77
distance of A(1,3) from BC is | (4.1 +7.3 -77)/sqrt(65) | =52/sqrt(65)

Answer 12

thankyou

Answer 13

welcome dear

Answer 14

i hope u understood each step and would be able to solve similar problem u face

Answer 15

i have come across this type of question alot and struggle everytime. this will definately help!