Articles by "midpoint"
Twitter Facebook

11:12 PM , , , ,

Problem: Given these pairs of points, (0, 9)) and (6, -9), find the midpoint, distance, slope, and equation of the line.

(0,9),(6,-9)\,

Solutions:

  • To find the midpoint, average the x coordinates and y coordinates. The midpoint is
\left(\frac{0+6}{2},\frac{9-9}{2}\right) = (3,0)\,
  • To find the (always zero or positive) distance, use the formula

d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,
d = \sqrt{(6)^2+(-9-9)^2} = \sqrt{36+18^2} = \sqrt{(2\cdot 3)^2+(2\cdot 3^2)^2} = \sqrt{(2\cdot 3)^2(1+3^2} = 6\sqrt{10}\,
d = \sqrt{(6)^2+(-9-9)^2} = \sqrt{36+18^2} = \sqrt{(2\cdot 3)^2+(2\cdot 3^2)^2} = \sqrt{(2\cdot 3)^2(1+3^2} = 6\sqrt{10}\,
  • To find the slope, use the formula
 m = \frac{y_2-y_1}{x_2-x_1}\,
m = \frac{-9-9}{6-0} = -3\,
The equations of the line are

Method 1:
 y=mx+b\,
  • Plug in one known point (say, (0, 9)) and the calculated slope.
9 = -3\cdot 0 + b\,
b = 9\,
  • Now plug b and m into the line equation:
  • y = -3x + 9\,
Method 2:
(y-y_1) = m(x-x_1)\,
  • Plug in one known point (say, (6, –9) ) and the calculated slope.
(y+9) = -3(x-6)\,
y = -3x + 18 - 9\,
  • y = -3x + 9\,

List of Similar Problems with Complete Solutions

NOTE:


For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

credit: Todd
Twitter Facebook

10:55 PM , , , ,

Problem: Given these pairs of points, (-5, 15) and (-5, 9), find the midpoint, distance, slope, and equation of the line.

(-5,15),(-5,9)\,

Solutions:

  • To find the midpoint, average the x coordinates and y coordinates. The midpoint is
\left(\frac{-5-5}{2},\frac{15+9}{2}\right) = \left(-5,\frac{15+9}{2}\right) = \left(-5,\frac{15+9}{2}\right)\,
  • To find the (always zero or positive) distance, use the formula

d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,
d = \sqrt{(-5+5)^2+(15-9)^2} = \sqrt{0^2+6^2} = \sqrt{6^2} = 6\,
  • To find the slope, use the formula
 m = \frac{y_2-y_1}{x_2-x_1}\,
m = \frac{9-15}{-5-(-5)} = undefined\, (which means it is a vertical line, which has infinite slope)
  • Since the line is vertical, there is only one x value that will give all y values. The equation for the line is
x = -5\,

    List of Similar Problems with Complete Solutions

    NOTE:


    For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

    credit: Todd©2013 www.FroydWess.com
    Twitter Facebook

    10:33 PM , , , ,

    Problem: Given these pairs of points, (4, 13) and (0, 9), find the midpoint, distance, slope, and equation of the line.

    (4,13),(09,0)\,

    Solutions:

    • To find the midpoint, average the x coordinates and y coordinates. The midpoint is
    \left(\frac{4+0}{2},\frac{13+9}{2}\right) = (2,11),
    • To find the (always zero or positive) distance, use the formula

     d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,
    d = \sqrt{(4-0)^2+(13-9)^2} = \sqrt{(4)^2+4^2} = \sqrt{16+16} = \sqrt{16*2} = \sqrt{4*4*2} = 4\sqrt{2}\,
    d = \sqrt{(4-0)^2+(13-9)^2} = \sqrt{(4)^2+4^2} = \sqrt{16+16} = \sqrt{16*2} = \sqrt{4*4*2} = 4\sqrt{2}\,
    • To find the slope, use the formula
     m = \frac{y_2-y_1}{x_2-x_1}\,
    m = \frac{9-13}{0-4} = \frac{-4}{-4} = 1,
    • The equations of the line are
    Method 1:
     y=mx+b\,
    Plug in one known point (say, (0, 9)) and the calculated slope.
    9 = 0 + b\,
    b = 9\,
    Now plug b and m into the line equation:
    • y = x + 9\,
    Method 2:
     (y-y_1) = m(x-x_1)\,

    Plug in one known point (say, (4, 13)) and the calculated slope.

    (y-13) = (x-4)\,
    y = x + 9\,
    • y = x + 9\,

    List of Similar Problems with Complete Solutions

    NOTE:


    For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

    credit: Todd
    Twitter Facebook

    10:04 PM , , , ,

    Problem: Given these pairs of points, (12, 1) and (4, 0), find the midpoint, distance, slope, and equation of the line.

    (12,1),(4,0)\,

    Solutions:

    • To find the midpoint, average the x coordinates and y coordinates. The midpoint is
    \left(\frac{12+4}{2},\frac{1+0}{2}\right) = \left(8,\frac{1}{2}\right)\,
    • To find the (always zero or positive) distance, use the formula
     d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,
    d = \sqrt{(12-4)^2+(1-0)^2} = \sqrt{(8)^2+1^2} = \sqrt{64+1} = \sqrt{5\cdot 15} = \sqrt{5\cdot 3\cdot 5} = 5\sqrt{3}\,
    d = \sqrt{(12-4)^2+(1-0)^2} = \sqrt{(8)^2+1^2} = \sqrt{64+1} = \sqrt{5\cdot 15} = \sqrt{5\cdot 3\cdot 5} = 5\sqrt{3}\,
    • To find the slope, use the formula
    m = \frac{y_2-y_1}{x_2-x_1}\,
    m = \frac{0-1}{4-12} = \frac{-1}{-8} = \frac{1}{8}\,
    • The equations of the line are
    Method 1:
     y=mx+b\,
    Plug in one known point (say, (4, 0) ) and the calculated slope.
    0 = \frac{1}{8}\cdot 4 + b\,
    b = -\frac{4}{8} = -\frac{1}{2}\,
    Now plug b and m into the line equation:
    • y = \frac{1}{8}x - \frac{1}{2}\,
    Method 2:
     (y-y_1) = m(x-x_1)\,
    Plug in one known point (say, (12, 1) ) and the calculated slope.
    (y-1) = \frac{1}{8}(x-12)\,
    y = \frac{1}{8}x - \frac{12}{8} + 1 = \frac{1}{8}x - \frac{4}{8} \,
    • y = \frac{1}{8}x - \frac{1}{2}\,

    List of Similar Problems with Complete Solutions

    NOTE:


    For any Questions, Clarifications and any violent Reactions use the comment section below. I'll be much willing to assist you.

    credit: Todd

    Contact Form

    Name

    Email *

    Message *

    Powered by Blogger.
    Javascript DisablePlease Enable Javascript To See All Widget