Algebra 1 help please

Question

Use formulas to complete the following problems and show all calculations.

1. Find M the midpoint between A and B given that a=(-1, 5) and b=(-3, -7)

2. Calculate the distance between A and B. Leave solution in simplest radical form.

3. Find the equation in slope-intercept form of the line that is the perpendicular bisector of the segment between (-3,2) and (3,-8)

4. In the box shown at the right, what is the length of the dashed line.

( file:///C:/Users/Speedy/Desktop/a00c48a9-7007-4095-86bc-c0a5c4ed38e3.jpg  )

Answer 1

1.  The midpoint formula:  the midpoint of a line segment with endpoints (x₁,y₁) and (x₂,y₂) is: 

     Midpoint:   (   (x₁+x₂)/2,  (y₁+y₂)/2  )

   That means that the x-coordinate of the midpoint is the average of the x-coordinates of the endpoints;  the y-coordinate of the midpoint is the average of the y-coordinates of the endpoints. 

       Midpoint:  (   (-1-3)/2,  (5-7)/2  )

       Midpoint:    ( -2, -1 )

    2. The distance formula:  the distance between two points, (x₁,y₁) and (x₂,y₂) can be expressed in terms of the two sides of the triangle that the points form with lines parallel to the x-axis and the y-axis (using the Pythagorean Theorem).  The distance formula is:

      D = √ ( (x₂-x₁)² + (y₂-y₁)²)

       D = √ (  (-3-(-1))² + (-7-5)²  )

      D = √ (  (-2)² + (-12)² ) 

     D = √ (  4 + 144  ) 

     D = 2 √(37) 

 3.  The slope of a line is often called “rise over run” because it is the increase in the y-coordinates divided by the increase in the x-coordinates.  Note:  positive slope means that the line points upward to the right (North-East); negative slope means that the line goes downward to the right (S-E).      The slope m =  (y₂-y₁) / (x₂-x₁) 

We have new points now:  (-3,2) and (3,-8) 
 
           m = (-8-2) / (3-(-3)) 

          m = -10 / (6) 

          m = =-5/3 

4.  The slope-intercept form of the equation of a line looks like this

:       y = mx + b       where m is slope   and b is y-intercept

      From (3) we have m=-5/3.

    All perpendicular lines have a slope that is the negative reciprocal of the slope of this line.  So, their m=3/5.

     Their equation looks like:

        y = (3/5)x + b             [and we need a point to determine b] 

   The perpendicular line goes through the midpoint (“perpendicular bisector”), so plug in (0,-3) -- you know how to find this]:

       -3 = (3/5)(0) + b 

      -3 = b

   The perpendicular bisector line has the equation:

       y = (3/5)x - 3
 
4.  I can’t see the jpg file on your desktop, but you should have enough info now to use the distance formula to determine the length of a line segment.

Answer 2

Hi Ayriel,

 

1) the midpoint formula is

 

x₁+x₂    y₁+y₂

------- , --------

    2          2

 

-1+(-3)    5+(-3)

--------  , --------

    2            2

 

(-4/2, 2/2)

(-2, 1)

 

2) The distance formula is

d = √[(x₁-x₂)2+(y₁-y₂)²]

 

d = √{[-1-(-3)]² + [5-(-7)]²}

d = √{[2]² + [12]²}

d = √{4 + 144} = √148 = 2√37

 

3) the slope intercept equation for the perpendicular bisector.

Given the original point (-1,5) and (-3,-7), we must first determine the slope

  -7-5           -12

--------  =  ------- = 6

-3-(-1)           -2

The slope of a perpendicular line would be -1/6 (the negative inverse)

This perpendicular line will go through the original line at a point which is the midpoint of (-3,2) and (3,-8)

The midpoint of these two points is

 

-3-3    2-(-8)

----- , -------

  2         2

 

(-6/2, 10/2) = (-3,5), using the approach given in 1).

 

So we need the equation of a line with a sloper of -1/6 which goes through the point (-3,5).

 

y-5 = (-1/6) (x+3)

y-5 = -1/6 x - 3/6

y-5 = -1/6 x - 1/2

y = -x/6 + 9/2    <----this is the slope-intercept form

 

4) I can't comment on this because your "link" is to a hard drive, not the internet.

 

OK, based on the photo you send, we have the following.  A rectangular prism which is 12cm x 9 cm and 8 cm high.

We want to find the length of the diagonal line from the bottom of the prism on one corner to the top of the prism on the other corner.

The first thing I would do is determine the diagonal of the top or bottom of the prism as presented.

 

We do that by using Pythagorean theorem.

a² + b² = c²

In this case a=9, b=12 and we want to determine c

81 + 144 = 225 = c²

c = 15

So imagine c as a dotted line diagonally across the bottom of the prism.

 

Now think of this as a triangle with the base being the c=15 we just calculated and the height is 8.  We want to determine the length of the original dashed line which is the hypotenuse of the triangle.

Using the same formula, we have

 

225 + 64 = 289 = (diagonal)²

diagonal = √289 = 17

 

Answer 3

1)  midpoint coordinates are (-2, 1)   ⇐ (average of x values of endpoints, average of y values of endpoints)

 

2) length AB = 2√37 using the Distance formula

 

3) line joining the two given points has midpoint M(0,-3) and has slope -5/3.  The perpendicular bisector of the segment passes through M and its slope is negative reciprocal of the -5/3, so, using slope 3/5 and point M in Point-Slope equation form, we have y -(-3) = (3/5)(x - 0). Then this final answer must be y = (3/5)x - 3