Online Notes and Lectures: midpoint

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Saturday, December 28, 2013

Given (0, 9) and (6, -9), find the midpoint, distance, slope, and equation of the line.

0 comments Posted by Anonymous at 11:12 PM

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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

$d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,$

$d = \sqrt{(6)^2+(-9-9)^2} = \sqrt{36+18^2} = \sqrt{(2\cdot 3)^2+(2\cdot 3^2)^2} = \sqrt{(2\cdot 3)^2(1+3^2} = 6\sqrt{10}\,$

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

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

Tags: distance, equation of line, Geometry Problems, midpoint, slope

Given (-5, 15) and (-5, 9), find the midpoint, distance, slope, and equation of the line.

0 comments Posted by Anonymous at 10:55 PM

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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.

Tags: distance, equation of line, Geometry Problems, midpoint, slope

Given (4, 13) and (0, 9), find the midpoint, distance, slope, and equation of the line.

0 comments Posted by Anonymous at 10:33 PM

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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

$d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,$

$d = \sqrt{(4-0)^2+(13-9)^2} = \sqrt{(4)^2+4^2} = \sqrt{16+16} = \sqrt{16*2} = \sqrt{4*4*2} = 4\sqrt{2}\,$

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.

Tags: distance, equation of line, Geometry Problems, midpoint, slope

Given (12, 1) and (4, 0), find the midpoint, distance, slope, and equation of the line.

0 comments Posted by Anonymous at 10:04 PM

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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

$d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,$

$d = \sqrt{(12-4)^2+(1-0)^2} = \sqrt{(8)^2+1^2} = \sqrt{64+1} = \sqrt{5\cdot 15} = \sqrt{5\cdot 3\cdot 5} = 5\sqrt{3}\,$

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.

Tags: distance, equation of line, Geometry Problems, midpoint, slope

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Saturday, December 28, 2013

Given (0, 9) and (6, -9), find the midpoint, distance, slope, and equation of the line.

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

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

Given (-5, 15) and (-5, 9), find the midpoint, distance, slope, and equation of the line.

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

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

Given (4, 13) and (0, 9), find the midpoint, distance, slope, and equation of the line.

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

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

Given (12, 1) and (4, 0), find the midpoint, distance, slope, and equation of the line.

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

Solutions:

List of Similar Problems with Complete Solutions

NOTE:

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