Equidistant
Equidistant is another word for 'equally distant', which means at the same distance from a place. A point is equidistant from other points if it is at the same distance away from them. Equidistant is a term that is mostly used in geometry in the concept of parallel lines, perpendicular bisectors, circles, angle bisectors, and so on.
| 1. | Equidistant Definition |
| 2. | What is the Equidistant Formula? |
| 3. | Solved Examples on Equidistant |
| 4. | Practice Questions on Equidistant |
| 5. | FAQs on Equidistant |
Equidistant Definition
A point is said to be equidistant from two other points when it is at an equal distance away from both of them. For example, the perpendicular bisector of a line segment is equidistant from both endpoints. Observe the following figure which shows that point Q is equidistant from P and R.
What is the Equidistant Formula?
There are two main formulas that are used under the topic of equidistant. We use the distance formula to find the distance between any two given points and the midpoint formula is used to find the midpoint of a line segment.
Distance Formula
The distance formula is used to find the distance between any two given points. We need to know the coordinates of the two points to use the distance formula. If the coordinates of the two points are: P \((x_{1},y_{1})\) and Q \((x_{2},y_{2})\) the distance, 'd' between these two points will be:
Midpoint Formula
A midpoint is a point that lies in the middle of two other points or in the middle of the line which joins any two points. To find the midpoint, we use the midpoint formula which is the average of the x-coordinates and the average of the y-coordinates of the two given points. Let \((x_{1},y_{1})\) and \((x_{2},y_{2})\) be the endpoints of any line segment. The midpoint formula for this line segment will be:
Important Notes
- Two or more objects or points are said to be equidistant if they are at the same distance from a place.
- A midpoint is also equidistant from the two original points.
- We can use either the midpoint formula or the distance formula to show that a point is equidistant from the two given points.
Related Articles on Equidistant
Check out the interesting articles mentioned below to grab more information about equidistant.
Solved Examples on Equidistant
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Example 2. The diameter of a circle has endpoints: (2, -3) and (-6, 5).
Can you find the coordinates of the center of this circle?
Solution:
The center of a circle is the midpoint of the diameter. Therefore, the coordinates of the center can be calculated using the midpoint formula:
Midpoint = \((x_{1} + x_{2})/2\) and \((y_{1} + y_{2})/2\) ; Here, \(x_{1}\) = -6, \(y_{1}\) = 5, \(x_{2}\) = 2, and \(y_{2}\) = -3
Substituting the values in the formula:
= [(-6 + 2)/2 , (5 + (-3))/2)
= (−4/2 , 2/2)
= (−2,1)
Therefore, the center of the circle is (-2,1). -
Example 3. Consider the line segment AB shown below.
The points A = (1, h) and B = (5, 7) are equidistant from their midpoint M. Find the value of 'h' if the midpoint of AB is M (3,−2).
Solution:
Here, \(x_{1}\) = 1, \(y_{1}\) = h, \(x_{2}\) = 5, and \(y_{2}\) = 7; Midpoint = (3,−2)
According to the midpoint formula: Midpoint =\((x_{1} + x_{2})/2\) and \((y_{1} + y_{2})/2\)
Substituting the values in the formula:
(3,−2) = [(1 + 5)/2 , (h + 7)/2]
(h + 7)/2 = −2
h + 7 = −4
h = −11
Therefore, the value of h = −11
FAQs on Equidistant
What is Equidistant in Geometry?
Equidistant means " a point which is at the same or equal distance from two given points." For example, the perpendicular bisector of a line segment is equidistant from both the endpoints.
How do you Find a Midpoint?
Midpoint is the average of the x-coordinates and the average of the y-coordinates of the two given points. When the coordinates of any two points A [\(x_{1}\) and \(y_{1}\))] and B [\(x_{2}\) and \(y_{2}\)] are given, the midpoint can be calculated using the formula: Midpoint =\((x_{1} + x_{2})/2\) and \((y_{1} + y_{2})/2\).
What is the Equidistant Formula?
The distance between any two given points can be calculated with the help of the distance formula: d = \(√[(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2]\); here [\(x_{1}\) and \(y_{1}\))] are the coordinates of one point and [\(x_{2}\) and \(y_{2}\)] are the coordinates of the other point. These coordinates are placed in the formula which gives the distance between the two points.
Is the Midpoint Between Two Given Points Always Equidistant from them?
Yes, since the midpoint of a line segment is at an equal distance from the two endpoints, it is said to be equidistant from them.
How do you know if a Point is Equidistant?
A point is said to be equidistant from two other points when it is at an equal distance away from both of them. The distance between any two given points can be calculated by using the distance formula with the help of the coordinates of the two points.