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Equidistant Formula
Equidistant means equal distance from every point. For example, consider the line segment containing the end points A and B and midpoint P. These points are said to be equidistant if the distance between the point A and P is equal to the distance between the point P and B, that means the point P is the mid point of A and B. Therefore, we can call equidistant as mid point. It is shown in below figure.
Equidistant formulas to find equidistant distance:
To find equidistant distance for any two end points, we have to use both mid point formula and distance formula. The midpoint formula and distance formula are given below.
Mid point formula = ((x_{1}+x_{2}) / 2, (y_{1}+y_{2}) / 2)
Distance formula = √((x_{2} – x_{1})^{2}+(y_{2}y_{1})^{2})
Example:
Find the equidistant for the points A(8, 8) and B(4, 8) using above formulas.
Solution:
Here, x_{1}= 8
x_{2} = 4
y_{1}= 8
y_{2} = 8
Step 1 : Find mid point for the AB
AB = ((x_{1}+x_{2} )/ 2,(y_{1}+y_{2}) / 2)
= ((8 + 4) / 2, (8 + 8) / 2)
= (12 / 2, 16 / 2)
Therefore, the mid point of AB = (6, 8)
Consider P as mid point of AB.
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Step 2: Find distance between A(8, 8) and C(6, 8) using distance formula.
AC = √(x_{2} – x_{1})^{2} + (y_{2}y_{1})^{2}
= √(6 – 8)^{2} + (8 – 8)^{2}
= √(2)^{2} + 0
= 2
Distance between AC = 2
Step 3: Find distance between C(6, 8) and B(4, 8) using distance formula.
AC = √(x_{2} – x_{1})^{2} + (y_{2}y_{1})^{2}
= √(4 – 6)^{2} + (8 – 8)^{2}
= √(2)^{2} + 0
= 2
Distance between CB = 2
Here,
AC = CB
Therefore, AC and CB are equidistant.
Example problem using equidistant formula :
Find x, if the point A(x, 6) is equidistant from C(6, 4) and D(14, 8)
Solution:
Step 1: Find the distance between the points A(x, 6) and C(6, 4).
Distance, AC = √ ((6 – x)^{2} + (4 6)^{2})
= √ ((36 + x^{2} – 12x) + (10)^{2} )
= √ (x^{2} – 12x + 136)
Step 2: Find the distance between the points A(x, 6) and D(14, 8).
Distance, A = √ ((14 – x)^{2} + (8 6)^{2})
= √ ((196 + x^{2} – 28x) + 2^{2} )
= √ (x^{2} – 28x + 200)
Step 3: AC = AD, since AC and AD are equidistant.
√ (x^{2} – 12x + 136) = √ (x^{2} – 28x + 200)
Squaring on both sides, we get
x^{2} – 12x + 136 = x^{2} – 28x + 200
– 12x + 28x = 200 – 136
16x = 64
x = 64 / 16
Therefore, x = 4
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