(3,4),(5,5) what is the distance and mid point
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Answer:
this is not the answer only explanation
Step-by-step explanation:
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Distance and midpoints
Midpoint formula
CCSS.Math: HSG.GPE.B.6
Walk through writing a general formula for the midpoint between two points.
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The \blueD{\text{midpoint}}midpointstart color #11accd, start text, m, i, d, p, o, i, n, t, end text, end color #11accd of the points (\greenD{x_1}, \goldD{y_1})(x
1
,y
1
)left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and (\greenD{x_2}, \goldD{y_2})(x
2
,y
2
)left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis is given by the following formula:
\left(\greenD{\dfrac{x_1+x_2}{2}}, \goldD{\dfrac{y_1+y_2}{2}}\right)(
2
x
1
+x
2
,
2
y
1
+y
2
)left parenthesis, start color #1fab54, start fraction, x, start subscript, 1, end subscript, plus, x, start subscript, 2, end subscript, divided by, 2, end fraction, end color #1fab54, comma, start color #e07d10, start fraction, y, start subscript, 1, end subscript, plus, y, start subscript, 2, end subscript, divided by, 2, end fraction, end color #e07d10, right parenthesis
In this article, we're going to derive this formula!
Deriving the midpoint formula
Let's start by plotting the points (\greenD{x_1}, \goldD{y_1})(x
1
,y
1
)left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and (\greenD{x_2}, \goldD{y_2})(x
2
,y
2
)left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis.
The \blueD{\text{midpoint}}midpointstart color #11accd, start text, m, i, d, p, o, i, n, t, end text, end color #11accd is the point halfway between each of the points:
An expression for the xxx-coordinate of the \blueD{\text{midpoint}}midpointstart color #11accd, start text, m, i, d, p, o, i, n, t, end text, end color #11accd is \greenD{\dfrac{x_1+x_2}{2}}
2
x
1
+x
2
start color #1fab54, start fraction, x, start subscript, 1, end subscript, plus, x, start subscript, 2, end subscript, divided by, 2, end fraction, end color #1fab54: Why does this expression work?
Similarly, an expression for the yyy-coordinate of the \blueD{\text{midpoint}}midpointstart color #11accd, start text, m, i, d, p, o, i, n, t, end text, end color #11accd is \goldD{\goldD{\dfrac{y_1+y_2}{2}}}
2
y
1
+y
2
start color #e07d10, start color #e07d10, start fraction, y, start subscript, 1, end subscript, plus, y, start subscript, 2, end subscript, divided by, 2, end fraction, end color #e07d10, end color #e07d10:
That's it! We derived the following formula for the \blueD{\text{midpoint}}midpointstart color #11accd, start text, m, i, d, p, o, i, n, t, end text, end color #11accd!
\left(\greenD{\dfrac{x_1+x_2}{2}}, \goldD{\dfrac{y_1+y_2}{2}}\right)(
2
x
1
+x
2
,
2
y
1
+y
2
)left parenthesis, start color #1fab54, start fraction, x, start subscript, 1, end subscript, plus, x, start subscript, 2, end subscript, divided by, 2, end fraction, end color #1fab54, comma, start color #e07d10, start fraction, y, start subscript, 1, end subscript, plus, y, start subscript, 2, end subscript, divided by, 2, end fraction, end color #e07d10, right parenthesis
Interestingly, a lot of people don't memorize this exact formula. Instead, they remember that to find the midpoint, you take the average of the xxx-coordinates and the average of the yyy-coordinates.