Post a New Question

Geometry

posted by .

Within an orthonormal system consider points A(1,4) B(5,1) C(4,8)

(a)Calculate the angle ABC

(b)Show that 3x+4y-19=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

  • Geometry -

    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
    (x-3)^2 +(y-2.5)^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) |BA||BC| sin ABC
    = (1/2)(5)(5 sqrt 2)(sin45)
    = (25/2) sqrt 2 * 1/sqrt 2
    = 25/2 = 12.5

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Math

    This question makes reference to an orthonormal basis (i,j), and an origin O. 1. Consider the triangle ABC, with vertices A(0,-2), B(9,1), C(-1,11). Find: a) a cartesian equation of the altitude from C; b) a cartesian equation of the …
  2. Math 101

    Find the equation for each of the following items below: a. A line that passes through (6,26) and has a slope of 3. b. A line that passes though the points (5,5) and (10,20) c. A line that passes through (9,25) and has a slope of -3. …
  3. math

    i have more than one question so if u no any of the answers please tell me 1.) write the point-slope form of the equation of the line with slope -2 passing through the point ( -5, -9). 2.) write the point-slope form of an equation …
  4. mat 101

    Equation for a line that passes through (6,26) and has a slope of 3 Equation for a line that passes through the points (5,5) and (10,20) Equation for a line that passes through (9,25) and has a slope of -3. Equation for a line that …
  5. Math - Quadratics & Analytical Geometry

    I have a test tomorrow with a few sample questions that are expected to be on the test that I do not know how to properly do. 1)Find the slope intercept equation of the line passing through the intersection of the lines 2x-4y=1 & 3x+4y=4 …
  6. Math -Quadratics Help!

    I have a test tomorrow with a few sample questions that are expected to be on the test that I do not know how to properly do. 1)Find the slope intercept equation of the line passing through the intersection of the lines 2x-4y=1 & 3x+4y=4 …
  7. Analytic Geometry

    A line Q has an equation of x+5y+5=0. Find the equation of the line that is "perpendicular" to Q an passes through (3,1). Find the equation of the line that is "parallel" to Q and passes to through (3,1). Find the equation of the line …
  8. Analytic Geometry

    A line Q has an equation of x+5x+5=0. Find the equation of the line that is "perpendicular" to Q and passes through (3,1). Find the question of the line that is "parallel" to Q and passes to through (3,1). Find the equation of the …
  9. Math

    What is the y-intercept of y = 2/3x + 2? I have to write it out but how?
  10. Math please.

    What is the y-intercept of y = 2/3x + 2? I have to write it out but how?

More Similar Questions

Post a New Question