# Kalkulator gradien, jarak dan titik tengah

> Cari jarak, titik tengah, gradien, persamaan garis dalam bentuk gradien-intersep, standar dan titik-gradien, serta perpotongan dua garis.

Versi interaktif: https://www.calcopenly.com/id/geometry/coordinate-geometry-calculator
Topik: Kalkulator geometri

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.

## Masukan

- **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**

## Hasil

- Distance P₁P₂ — hasil utama
- 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

## Rumus

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

## Contoh penyelesaian

### 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**
- Sumber pemeriksaan: 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**
- Sumber pemeriksaan: 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**
- Sumber pemeriksaan: 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: ya
- 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**
- Sumber pemeriksaan: 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: ya
- 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**
- Sumber pemeriksaan: 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: ya
- 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 °**
- Sumber pemeriksaan: Both slopes are 1 with different intercepts (0 and 1)

## Pertanyaan

### 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.

### Seberapa akurat “Kalkulator gradien, jarak dan titik tengah”?

Akurasi bergantung pada masukan dan asumsi metode. Aritmetika desimal memakai 50 digit signifikan, tetapi perkiraan, metode numerik dan data sumber bisa kurang presisi; pembulatan yang ditampilkan tidak menghilangkan batasan itu. Contoh penyelesaian yang diperiksa dengan sumber independen: 6. Misalnya, “P₁(1, 2) and P₂(4, 6)” diperiksa dengan 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)).

### Dari mana metode ini berasal?

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

## Sumber

- [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)