How to Use "Equidistant" and "Equidistance" With Examples
Understanding Equidistant is crucial in geometry, board exams, and competitive tests. The idea of equal distance helps solve problems about circles, triangles, maps, and more. Knowing this concept makes it easier to analyze shapes and distances in both maths and real-life scenarios.
Formula Used in Equidistant
The standard formula is: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), where 'd' is the distance between two points. The midpoint formula, used for finding a point equidistant from both ends of a segment, is: \( \Big(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\Big) \).
Here’s a helpful table to understand Equidistant more clearly:
Equidistant Table
| Word | Value | Applies? |
|---|---|---|
| Midpoint | (x₁ + x₂)/2, (y₁ + y₂)/2 | Yes |
| Distance | √[(x₂-x₁)² + (y₂-y₁)²] | Yes |
| Random Point | (3,5) | No |
This table shows how the pattern of Equidistant appears with formulas for finding middle points or distances in geometry.
Worked Example – Solving a Problem
Let’s solve: Find a point C on the line segment AB, where A(2, 1) and B(8, 5), so that C is equidistant from both A and B (that is, the midpoint).
1. Write the midpoint formula: $\Big(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\Big)$2. Substitute values: $x_1 = 2, y_1 = 1, x_2 = 8, y_2 = 5$
3. Calculate x-coordinate: \( \frac{2 + 8}{2} = \frac{10}{2} = 5 \)
4. Calculate y-coordinate: \( \frac{1 + 5}{2} = \frac{6}{2} = 3 \)
5. Final answer: The point C(5, 3) is equidistant from A and B.
You can see the same concept of equidistance in the perpendicular bisector or the mid-point theorem, which are vital in triangle and coordinate geometry proofs.
Practice Problems
- Find a point equidistant from (4, 2) and (10, 8).
- Is the midpoint always equidistant from the endpoints? Explain why.
- On a circle with center (0, 0), show that any point (x, y) on the circle is equidistant from the center.
- List three real-life examples of equidistant points or places.
Common Mistakes to Avoid
- Confusing Equidistant with congruent shapes (having the same size and shape, not distance).
- Forgetting that only points on a perpendicular bisector are equidistant from the segment’s endpoints.
- Assuming a random point between two others must always be equidistant—only the midpoint is.
Real-World Applications
Equidistant concepts are everywhere: city planning (placing stations at equal distances), robotics (moving along the center between lines), and in maps, where we use locus of points. In a triangle, the circumcenter is equidistant from all three vertices. Vedantu brings these ideas to life with practical examples.
We explored the idea of Equidistant, how to find midpoints, use distance in geometry, and apply it to real-world problems. Practice more with Vedantu to master geometry and spot patterns of equidistance in maths and beyond.
Related: The locus of equidistant points forms the basis for constructions and proofs, while the perpendicular bisector is a key geometric concept you will use often.
FAQs on What Does It Mean If Something Is Equidistant?
1. What does it mean if something is equidistant?
Equidistant means that a point or a set of points are at the same distance from two or more other points or objects. For example, if a point is equidistant from points A and B, it is located where the distances to A and B are equal.
2. What is another word for equidistant?
Common synonyms for equidistant include midway, centered, evenly spaced, or symmetrically distant. These words describe something located at an equal distance from two or more objects.
3. What is the meaning of equidistant in geometry?
In geometry, equidistant describes a point, line, or object that is at an equal distance from two or more specific points, lines, or planes. For example, any point on the perpendicular bisector of a segment is equidistant from the segment's endpoints.
4. How do you pronounce equidistant?
The word equidistant is pronounced as /ˌekwɪˈdɪstənt/ (ek-wi-DIS-tuhnt).
5. What does equidistant mean in math?
In math, especially in geometry and coordinate systems, equidistant means being at the same distance from two or more points or objects. For example, the locus of points equidistant from a point and a line forms a parabola.
6. Can you provide an example of equidistant in a sentence?
Here is an example sentence: "The school is equidistant from Ravi's and Anu's houses, making it a fair meeting point for both."
7. What is the equidistant formula?
The basic equidistant formula in coordinate geometry is: Distance from point P to A = Distance from point P to B. If P(x, y) is equidistant from A(x1, y1) and B(x2, y2), then:
√[(x - x1)² + (y - y1)²] = √[(x - x2)² + (y - y2)²]
8. What is an equidistant map projection?
An equidistant map projection is a type of map projection where distances from the center of the map to any other place are shown accurately to scale. Popular types include the Azimuthal Equidistant Projection, often used for aviation and radio mapping.
9. What does equidistantly mean?
The adverb equidistantly means "in an equidistant manner" or "at equal distances." For example, trees planted equidistantly form a straight, evenly spaced row.
10. What is the equidistant letter sequence?
The equidistant letter sequence (ELS) refers to a method of selecting letters from a text at regular intervals to form new words or patterns. This technique is most commonly discussed in relation to cryptography and biblical codes.
11. What is a synonym for equidistant?
A synonym for equidistant is midway, centered, or equally distant.
12. What is equidistance in geometry?
In geometry, equidistance refers to the property of being at equal distances from two or more objects. For example, all points on the perpendicular bisector of a line segment demonstrate equidistance from the segment’s endpoints.
