Analytic geometry: points and lines · Grade 10 · 4 units
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
For example: The points are A(−6, 8), B(−6, −7) . What is the distance between them?
Answer and solution
The answer: 15
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
For example: The points are A(−6, 8), B(−6, 0) . What are the coordinates of the midpoint of the segment joining them?
Answer and solution
The answer: (−6, 4)
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
For example: A line goes through the points A(−6, 7), B(−9, −2) . What is the gradient of the line?
Answer and solution
The answer: 3
The gradient is (−2 − 7) ÷ (−9 − (−6)) , that is −9 ÷ (−3) = 3 .
The equation of a line
For example: A line has a gradient of −4 and crosses the y axis at (0, 8) . What is the equation of the line?
Answer and solution
The answer: −4x + 8
The equation of the line is y = −4x + 8 .
Parallel and perpendicular lines
For example: Here are the lines y = −2x + 6, y = −2x . Which is true of the two lines?
- They are parallel
- They are the same line
- They intersect but are not perpendicular
- They are perpendicular
Answer and solution
The answer: They are parallel
The gradients are −2 and −2 . The gradients are equal and the constants differ, so the lines are parallel.
Points of intersection
For example: Here is the line y = −3x − 12 . At which point does it meet the x axis?
Answer and solution
The answer: (−4, 0)
Substitute zero for y: 0 = −3x − 12 , so x = 12 ÷ (−3) = −4 , and the point is (−4, 0) .
Lines parallel to the axes
For example: Which equation describes the line through the points A(−5, 8), B(−5, −6) ?
- x = 8
- y = −5
- y = −6
- x = −5
Answer and solution
The answer: x equals minus five
Both points have the same x coordinate, so the line is parallel to the y axis: x = −5 .
Shapes on a coordinate grid
For example: The triangle has vertices A(−5, 6), B(−3, 6), C(0, 7) . Find its area.
Answer and solution
The answer: 1
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
Parts of this topic that are not built 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, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).