# Analytic geometry: points and lines · Grade 10 · 4 units

> Explanation, worked examples and free practice following the Israeli curriculum.

In Analytic geometry: points and lines (Grade 10 · 4 units) you practise 8 skills: The distance between two points · The midpoint of a segment · The gradient of a line · The equation of a line · Parallel and perpendicular lines · Points of intersection · Lines parallel to the axes · Shapes on a coordinate grid.

The distance between two points on a coordinate grid is the hypotenuse of a right triangle: its legs are the difference of the x coordinates and the difference of the y coordinates, so the distance is the square root of the sum of the squared differences. The midpoint of a segment has coordinates that are the means of the endpoints’ coordinates, and when you know one endpoint and the midpoint, the other endpoint is just as far from the midpoint on the other side.

The gradient of a line is the change in y divided by the change in x between two points on it. A rising line has a positive gradient, a falling line a negative one, a line parallel to the x axis a gradient of zero, and a line parallel to the y axis has no gradient. The equation of a line in explicit form is y equals the gradient times x plus the y intercept; you find it from a gradient and a point, or from two points.

Two lines are parallel when their gradients are equal, which is the parallel axiom in coordinates. Two lines are perpendicular when the product of their gradients is minus one. The point where two lines meet solves both their equations at once, and a line meets an axis where the other variable is zero.

**Tip:** Sketch the points before you calculate: a small sketch shows whether the gradient is positive or negative and whether the answer is sensible.

## What you learn

### The distance between two points

- Example: The points are A(−6, 8), B(−6, −7) . What is the distance between them? (Two points on a grid, and the segment joining them.)
- Answer: 15
- Solution: The segment is parallel to an axis, so the distance is just the difference of the coordinates that change: AB = 15 .

### The midpoint of a segment

- Example: The points are A(−6, 8), B(−6, 0) . What are the coordinates of the midpoint of the segment joining them? (A segment on a grid, with its endpoints marked.)
- Answer: (−6, 4)
- Solution: The x coordinate of the midpoint is (−6 + (−6)) ÷ 2 = −6 , and its y coordinate is (8 + 0) ÷ 2 = 4 . So M(−6, 4) .

### The gradient of a line

- Example: A line goes through the points A(−6, 7), B(−9, −2) . What is the gradient of the line? (Two points on a grid, and the line through them.)
- Answer: 3
- Solution: The gradient is (−2 − 7) ÷ (−9 − (−6)) , that is −9 ÷ (−3) = 3 .

### The equation of a line

- Example: A line has a gradient of −4 and crosses the y axis at (0, 8) . What is the equation of the line?
- Answer: −4x + 8
- Solution: The equation of the line is y = −4x + 8 .

### Parallel and perpendicular lines

- Example: Here are the lines y = −2x + 6, y = −2x . Which is true of the two lines?
- Options: They are parallel · They are the same line · They intersect but are not perpendicular · They are perpendicular
- Answer: They are parallel
- Solution: The gradients are −2 and −2 . The gradients are equal and the constants differ, so the lines are parallel.

### Points of intersection

- Example: Here is the line y = −3x − 12 . At which point does it meet the x axis?
- Answer: (−4, 0)
- Solution: Substitute zero for y: 0 = −3x − 12 , so x = 12 ÷ (−3) = −4 , and the point is (−4, 0) .

### Lines parallel to the axes

- Example: Which equation describes the line through the points A(−5, 8), B(−5, −6) ? (Two points on a grid, and the line through them.)
- Options: x equals eight x = 8 · y equals minus five y = −5 · y equals minus six y = −6 · x equals minus five x = −5
- Answer: x equals minus five
- Solution: Both points have the same x coordinate, so the line is parallel to the y axis: x = −5 .

### Shapes on a coordinate grid

- Example: The triangle has vertices A(−5, 6), B(−3, 6), C(0, 7) . Find its area. (A triangle on a grid, with one side parallel to the x axis.)
- Answer: 1
- Solution: The base AB = −3 − (−5) = 2 and the height 7 − 6 = 1 so the area is: 2 × 1 ÷ 2 = 1

## Not on the site yet

- Proofs with coordinates
- The equation of a circle
- Drawing a line on paper

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 10 · 4 units topics](https://kixplo.com/il/en/math/grade-10-4u/)