Polygons on the coordinate plane · Grade 12 · 3 units
In Polygons on the coordinate plane (Grade 12 · 3 units) you practise 5 skills: The gradient of a line · Triangles from their vertices · Areas · A polygon from the equations of its sides · Quadrilaterals from their vertices. When the vertices of a polygon are given on a grid, the coordinates tell you almost everything: the lengths of the sides, the perimeter, the area, even the kind of polygon. The length of a segment comes from Pythagoras, using the difference in the x coordinates and the difference in the y coordinates.
The gradient of a line is the change in y divided by the change in x. Lines are parallel when their gradients are equal, and perpendicular when the gradients multiply to minus one. That is how you recognise a parallelogram, a rectangle or a trapezium, and check whether a triangle has a right angle. For an area, choose a side parallel to one of the axes: its length and the height to it are differences of a single coordinate. When the sides are given as equations, each vertex is where two of the lines meet.
Tip: Plot the points on a grid before calculating. The sketch shows which side is parallel to an axis and where to start.
What you learn
The gradient of a line
For example: A line goes through the points A(7, −5), B(10, 1) . What is the gradient of the line?
Answer and solution
The answer: 2
The gradient is (1 − (−5)) ÷ (10 − 7) , that is 6 ÷ 3 = 2 .
Triangles from their vertices
For example: The vertices of the triangle are A(2, −7) , B(5, −3) and C(2, 1) .
What is the perimeter of the triangle?Answer and solution
The answer: 18
The squares of the sides, by Pythagoras: AB² = 3² + 4² = 25 , so AB = 5 , BC² = 3² + 4² = 25 , so BC = 5 , AC = 8 . The perimeter is 5 + 5 + 8 = 18 .
Areas
For example: The vertices of the triangle ABC are A(−3, 5) , B(−1, 8) and C(−6, 8) .
What is the area of the triangle?Answer and solution
The answer: 7.5
The side BC is parallel to the x axis, and its length is 5 . The height to it is |5 − 8| = 3 . The area is 5 × 3 ÷ 2 = 7.5 .
A polygon from the equations of its sides
For example: The line 4x − 3y = 12 and the two axes form a triangle. What is the perimeter of the triangle?
Answer and solution
The answer: 12
Substitute y = 0 to get x = 3 , and substitute x = 0 to get y = −4 . The vertices of the triangle are (0, 0), (3, 0), (0, −4) . The legs are 3 and 4 , and the hypotenuse by Pythagoras: 3² + 4² = 25 , so it is 5 . The perimeter is 3 + 4 + 5 = 12 .
Quadrilaterals from their vertices
For example: The vertices of the quadrilateral ABCD are A(−2, 2) , B(−4, −4) , C(1, −6) and D(3, 0) .
What is the most precise name for the quadrilateral?- A rectangle
- A trapezium
- None of these
- A parallelogram
Answer and solution
The answer: A parallelogram
The gradients of the sides, in order: m₁ = 3 , m₂ = −2/5 , m₃ = 3 and m₄ = −2/5 . Both pairs of opposite sides are parallel: m₁ = m₃, m₂ = m₄ . The gradients of two adjacent sides multiply to m₁ × m₂ = −6/5 , not −1 . So there is no right angle, and the quadrilateral is a parallelogram.
Not on the site yet
Parts of this topic that are not built yet: Analytic proofs · The equation of a circle · The area of a general polygon.
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 3 units (grades 10 and 11).