# ڈھلوان، فاصلہ اور وسطی نقطہ کیلکولیٹر

> دو نقطوں کا فاصلہ، وسطی نقطہ اور ڈھلوان نکالیں۔ خط کی مساوات ڈھلوان-قطع، معیاری اور نقطہ-ڈھلوان صورتوں میں، اور دو خطوں کا نقطۂ تقاطع دیکھیں۔

انٹرایکٹو صورت: https://www.calcopenly.com/ur/geometry/coordinate-geometry-calculator
موضوع: ہندسی کیلکولیٹر

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.

## اندراجات

- **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**

## نتائج

- Distance P₁P₂ — بنیادی نتیجہ
- 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

## فارمولا

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

## حل شدہ مثالیں

### 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**
- جانچ کا ماخذ: ⁨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**
- جانچ کا ماخذ: ⁨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**
- جانچ کا ماخذ: ⁨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: ہاں
- 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**
- جانچ کا ماخذ: ⁨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: ہاں
- 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**
- جانچ کا ماخذ: ⁨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: ہاں
- 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 °**
- جانچ کا ماخذ: ⁨Both slopes are 1 with different intercepts (0 and 1)⁩

## سوالات

### 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.

### “⁨ڈھلوان، فاصلہ اور وسطی نقطہ کیلکولیٹر⁩” کتنا درست ہے؟

درستی آپ کی درج کردہ قدروں اور طریقے کے مفروضوں پر منحصر ہے۔ اعشاری حساب 50 بامعنی ہندسے استعمال کرتا ہے، مگر تخمینے، عددی طریقے اور ماخذ کا ڈیٹا کم درست ہو سکتے ہیں؛ دکھائی گئی قدروں کو راؤنڈ کرنے سے یہ حدود ختم نہیں ہوتیں۔ آزاد ذرائع کی حل شدہ مثالوں سے جانچ: 6۔ مثلاً، “⁨P₁(1, 2) and P₂(4, 6)⁩” کو ⁨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))⁩ سے جانچا جاتا ہے۔

### اس طریقے کا ماخذ کیا ہے؟

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

## ماخذ

- [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)