# ಇಳಿಜಾರು, ದೂರ ಮತ್ತು ಮಧ್ಯಬಿಂದು ಕ್ಯಾಲ್ಕುಲೇಟರ್

> ಎರಡು ಬಿಂದುಗಳ ದೂರ, ಮಧ್ಯಬಿಂದು, ಇಳಿಜಾರು ಮತ್ತು ರೇಖೆಯ ಸಮೀಕರಣವನ್ನು ಇಳಿಜಾರು-ಛೇದ, ಪ್ರಮಾಣಿತ ಹಾಗೂ ಬಿಂದು-ಇಳಿಜಾರು ರೂಪಗಳಲ್ಲಿ ನೋಡಿ. ಎರಡು ರೇಖೆಗಳ ಛೇದನಬಿಂದುವನ್ನೂ ಲೆಕ್ಕಿಸಿ.

ಸಂವಾದಾತ್ಮಕ ಆವೃತ್ತಿ: https://www.calcopenly.com/kn/geometry/coordinate-geometry-calculator
ವಿಷಯ: ರೇಖಾಗಣಿತದ ಕ್ಯಾಲ್ಕುಲೇಟರ್‌ಗಳು

Enter two points to get the straight-line distance d = √((x₂ − x₁)² + (y₂ − y₁)²), the midpoint ((x₁ + x₂)/2, (y₁ + y₂)/2) and the slope m = (y₂ − y₁)/(x₂ − x₁). The line through them is written in slope-intercept form y = mx + b, in standard form Ax + By = C with whole-number coefficients and in point-slope form, alongside its perpendicular bisector. Fractions stay exact, so a slope of 4/3 is not rounded to 1.3333.

The default points (1, 2) and (4, 6) are 5 apart, a 3-4-5 triangle, with midpoint (2.5, 4) and line y = (4/3)x + 2/3. Switching on a second line adds whether the two are parallel, perpendicular or intersecting, the angle between them and the crossing point.

A vertical line has no slope, so only its standard form, such as x = 3, is given. Coordinates carry no unit: the distance is in whatever unit they use.

## ಒಳಾಂಶಗಳು

- **Point 1: x**
- **Point 1: y**
- **Point 2: x**
- **Point 2: y**
- **Compare with a second line**
- **Line 2, point 3: x**
- **Line 2, point 3: y**
- **Line 2, point 4: x**
- **Line 2, point 4: y**

## ಫಲಿತಾಂಶಗಳು

- Distance P₁P₂ — ಮುಖ್ಯ ಫಲಿತಾಂಶ
- Midpoint x
- Midpoint y
- Slope
- Angle of inclination (°)
- Slope-intercept form
- Standard form
- Point-slope form
- Perpendicular bisector
- Line 2
- The lines are
- Angle between the lines (°)
- Intersection x
- Intersection y

## ಸೂತ್ರ

$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},\quad M = \left(\tfrac{x_1 + x_2}{2}, \tfrac{y_1 + y_2}{2}\right),\quad m = \frac{y_2 - y_1}{x_2 - x_1},\quad \tan\theta = \left|\frac{m_2 - m_1}{1 + m_1 m_2}\right|
$$

## ಪರಿಹರಿಸಿದ ಉದಾಹರಣೆಗಳು

### P₁(1, 2) and P₂(4, 6)

- Point 1: x: 1
- Point 1: y: 2
- Point 2: x: 4
- Point 2: y: 6
- **Distance P₁P₂: 5**
- **Midpoint x: 2.5**
- **Midpoint y: 4**
- **Slope: 1.33333333**
- **Angle of inclination: 53.130102 °**
- **Slope-intercept form: y = (4/3)x + 2/3**
- **Standard form: 4x − 3y = −2**
- **Point-slope form: y − 2 = (4/3)(x − 1)**
- **Perpendicular bisector: y = −0.75x + 5.875**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: Python 3.8 fractions: dx = 3, dy = 4 (3-4-5), m = 4/3, b = 2 − 4/3 = 2/3; bisector through (5/2, 4) with slope −3/4 has b = 47/8 = 5.875 (terminating fractions print as decimals); math.degrees(atan(4/3))

### Vertical line (edge case: no slope)

- Point 1: x: 3
- Point 1: y: 1
- Point 2: x: 3
- Point 2: y: 7
- **Distance P₁P₂: 6**
- **Standard form: x = 3**
- **Angle of inclination: 90 °**
- **Perpendicular bisector: y = 4**
- **Midpoint y: 4**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: Equal x-coordinates: the line x = 3 is vertical; the bisector is the horizontal line through (3, 4)

### Decimal coordinates

- Point 1: x: -1.5
- Point 1: y: 2.25
- Point 2: x: 3.5
- Point 2: y: -0.75
- **Distance P₁P₂: 5.83095189**
- **Slope: -0.6**
- **Slope-intercept form: y = −0.6x + 1.35**
- **Standard form: 12x + 20y = 27**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: Python 3.8 fractions: dx = 5, dy = −3, √34; −3x − 5y = −27/4 scaled by −4

### Two intersecting lines

- Point 1: x: 1
- Point 1: y: 2
- Point 2: x: 4
- Point 2: y: 6
- Compare with a second line: ಹೌದು
- Line 2, point 3: x: 0
- Line 2, point 3: y: 6
- Line 2, point 4: x: 6
- Line 2, point 4: y: 0
- **The lines are: Intersecting**
- **Angle between the lines: 81.869898 °**
- **Intersection x: 2.28571429**
- **Intersection y: 3.71428571**
- **Line 2: y = −x + 6**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: tan θ = |(−1 − 4/3)/(1 − 4/3)| = 7, Python 3.8 math.degrees(atan(7)); intersection (16/7, 26/7) by Python fractions

### Perpendicular lines

- Point 1: x: 0
- Point 1: y: 0
- Point 2: x: 2
- Point 2: y: 1
- Compare with a second line: ಹೌದು
- Line 2, point 3: x: 0
- Line 2, point 3: y: 5
- Line 2, point 4: x: 1
- Line 2, point 4: y: 3
- **The lines are: Perpendicular**
- **Angle between the lines: 90 °**
- **Intersection x: 2**
- **Intersection y: 1**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: Slopes 1/2 and −2 multiply to −1; y = x/2 meets y = 5 − 2x at x = 2

### Parallel lines (edge case: no intersection)

- Point 1: x: 0
- Point 1: y: 0
- Point 2: x: 1
- Point 2: y: 1
- Compare with a second line: ಹೌದು
- Line 2, point 3: x: 0
- Line 2, point 3: y: 1
- Line 2, point 4: x: 2
- Line 2, point 4: y: 3
- **The lines are: Parallel**
- **Angle between the lines: 0 °**
- ಪರಿಶೀಲನೆಯ ಮೂಲ: Both slopes are 1 with different intercepts (0 and 1)

## ಪ್ರಶ್ನೆಗಳು

### How do you find the slope between two points?

Divide the change in y by the change in x: m = (y₂ − y₁)/(x₂ − x₁). From (1, 2) to (4, 6) the rise is 4 and the run is 3, so m = 4/3 ≈ 1.3333. A positive slope rises to the right, a negative one falls, 0 is horizontal, and when x₂ = x₁ the line is vertical and the slope is undefined.

### What is the distance formula?

d = √((x₂ − x₁)² + (y₂ − y₁)²), which is Pythagoras' theorem applied to the horizontal and vertical gaps. Between (1, 2) and (4, 6) the gaps are 3 and 4, so d = √25 = 5. For points given as latitude and longitude use a great-circle formula instead, because the flat formula ignores the Earth's curvature.

### How do you find the equation of a line through two points?

Find the slope m, then the intercept b = y₁ − m·x₁, and write y = mx + b. Through (1, 2) and (4, 6), m = 4/3 and b = 2 − 4/3 = 2/3, so y = (4/3)x + 2/3. Multiplying by 3 and rearranging gives the standard form 4x − 3y = −2, with whole-number coefficients.

### How do you tell if two lines are parallel or perpendicular?

Compare their slopes. Parallel lines have equal slopes, and perpendicular lines have slopes whose product is −1, such as 1/2 and −2. Any other pair crosses at an angle θ with tan θ = |(m₂ − m₁)/(1 + m₁m₂)|: slopes of 4/3 and −1 give tan θ = 7, so the lines meet at 81.87°.

### What is a perpendicular bisector?

The line through the midpoint of a segment at right angles to it; every point on it is equally far from both endpoints. For (1, 2) and (4, 6) it passes through (2.5, 4) with slope −3/4, the negative reciprocal of 4/3, giving y = −0.75x + 5.875. The bisectors of a triangle's three sides meet at the centre of its circumscribed circle.

### “ಇಳಿಜಾರು, ದೂರ ಮತ್ತು ಮಧ್ಯಬಿಂದು ಕ್ಯಾಲ್ಕುಲೇಟರ್” ಎಷ್ಟು ನಿಖರವಾಗಿದೆ?

ನಿಖರತೆ ನಿಮ್ಮ ಇನ್‌ಪುಟ್‌ಗಳು ಮತ್ತು ವಿಧಾನದ ಊಹೆಗಳನ್ನು ಅವಲಂಬಿಸಿದೆ. ದಶಮಾಂಶ ಗಣನೆ 50 ಸಾರ್ಥಕ ಅಂಕೆಗಳನ್ನು ಬಳಸುತ್ತದೆ. ಆದರೆ ಅಂದಾಜುಗಳು, ಸಂಖ್ಯಾತ್ಮಕ ವಿಧಾನಗಳು ಮತ್ತು ಮೂಲ ದತ್ತಾಂಶ ಕಡಿಮೆ ನಿಖರವಾಗಿರಬಹುದು; ಪ್ರದರ್ಶಿತ ಮೌಲ್ಯಗಳನ್ನು ರೌಂಡ್ ಮಾಡುವುದರಿಂದ ಈ ಮಿತಿಗಳು ನಿವಾರಣೆಯಾಗುವುದಿಲ್ಲ. ಸ್ವತಂತ್ರ ಮೂಲಗಳ ಪರಿಹಾರಗಳೊಂದಿಗೆ ಪರಿಶೀಲಿಸಿದ ಉದಾಹರಣೆಗಳು: 6. ಉದಾಹರಣೆಗೆ, “P₁(1, 2) and P₂(4, 6)” ಅನ್ನು Python 3.8 fractions: dx = 3, dy = 4 (3-4-5), m = 4/3, b = 2 − 4/3 = 2/3; bisector through (5/2, 4) with slope −3/4 has b = 47/8 = 5.875 (terminating fractions print as decimals); math.degrees(atan(4/3)) ಜೊತೆಗೆ ಪರಿಶೀಲಿಸಲಾಗುತ್ತದೆ.

### ಈ ವಿಧಾನದ ಮೂಲ ಯಾವುದು?

OpenStax College Algebra 2e, §2.1 (distance and midpoint formulas) and §2.2 (equations of lines); Weisstein, E. W. “Line”, “Perpendicular Bisector” — MathWorld.

## ಮೂಲಗಳು

- [OpenStax College Algebra 2e, §2.1 (distance and midpoint formulas) and §2.2 (equations of lines)](https://openstax.org/books/college-algebra-2e/pages/2-1-the-rectangular-coordinate-systems-and-graphs)
- [Weisstein, E. W. “Line”, “Perpendicular Bisector” — MathWorld](https://mathworld.wolfram.com/PerpendicularBisector.html)