Geometry
posted by Anonymous on .
Within an orthonormal system consider points A(1,4) B(5,1) C(4,8)
(a)Calculate the angle ABC
(b)Show that 3x+4y19=0 is an equation for line L which passes through A and B
(c)Find a vector equation for line L
(d)(i)give an equation for the circle with the line segment AB as diameter
(ii)Give an equation for the tangent to the circle at the point B
(e)(i)Find the distance from O,the origin,to line L
(ii)Give an equation for the line through O and perpendicular to L
(iii)Find an equation for the smallest circle through circle O which has a segment of L as a diameter
(f)Calculate the area of the triangle
ABC
I need mostly need help on c,d,e

Vector BA = 4i + 3j
Vector BC = 1i + 7j
BA = sqrt (16+9) = 5
BC = sqrt (1 + 49) = sqrt (50)=5sqrt2
BA dot BC = 4+21 = 25
BA dot BC = BA BC cos ABC
so
25 = 5 * 5 sqrt 2 * cos ABC
cos ABC = sqrt 2 / 3 = .707
ABC = 45 degrees
b
L is y = (3/4) x + 19/4
note slope of vector AB = 3/4
now does L go through A which is (1,4)?
4 = (3/4)x + 19/4
16 = 3x + 19
x = 1 sure enough L goes through A and B
c
did that right off in A but I did BA = 4i + 3 j
we want AB = 4i 3j = BA
d
center of circle at center of AB
x = (5+1)/2 = 3
y = (4+1)/2 = 2.5
radius = BA/2 = 2.5
so
(x3)^2 +(y2.5)^2 = 2.5^2 = 6.25
tangent at B has slope perpendicular to L 1/(3/4) or 4/3 and goes through B
y = (4/3) x + b
1 = (4/3)5 + b
b = 1  20/3 = 17/3
so y = (4/3) x 17/3
or 3 y = 4 x  17
e
call line perpendicular to L through the origin P
equation of P is
y = m' x + 0 (because through origin b =0)
m' = 1/slope of L = 4/3
so
y = (4/3)x is equation of P (that is e ii by the way
where does L hit P ?
(4/3) x = (3/4) x + 19/4
(4/3 + 3/4) x = 19/4
(16 + 9 ) x/12 = 19/4
25 x = 3*19
x = 3*19/25
y = 4*19/25
the distance from O to that intersection is (19/25)sqrt (3^2+4^2) =(19/25)*5 = 19/5 that is e i answer
e iii
well I know center is at x = 3*19/25 and y = 4*19/25 and the radius is 19/5
so
(x  3*19/25)^2 + (y  4*19/25)^2 = 19^2/25^2
f
area = (1/2) BABC sin ABC
= (1/2)(5)(5 sqrt 2)(sin45)
= (25/2) sqrt 2 * 1/sqrt 2
= 25/2 = 12.5