☰ خطة العام · الرياضيات (بالإنجليزية)

Distances, routes and shapes on the plane

الأسبوع 25 · المسافات على المستوى الإحداثي ورسم الأشكال من نقاط · 3 دروس
3

Drawing rectangles, squares and triangles from coordinates

Discover Mathematics P6: Term 2, Unit 11, Lessons 4–6
GoalFind a missing vertex and the side lengths of shapes drawn from coordinates.
New words
vertex · a corner of a shaperight angle · a square corner, made by a horizontal and a vertical linepolygon · a closed shape with straight sides
Explanation

On a coordinate plane a horizontal line and a vertical line meet at a right angle. So a rectangle with sides along the grid lines has pairs of vertices that share an x or a y. To find the missing vertex, take the x from the vertex in the same column and the y from the vertex in the same row. For A(−3, −1), B(4, −1), C(4, 3), the fourth vertex D shares x with A and y with C: D = (−3, 3). A square has equal sides: if one side is 4 units, all are 4. Side lengths come from distances along the axes. A rectangle 7 units by 4 has perimeter 2 × (7 + 4) = 22 and area 7 × 4 = 28 square units.

★Remember
The missing vertex copies x from the vertex in its column and y from the vertex in its row. Find side lengths with distances along the axes.
Worked examples
xy−6−5−4−3−2−1123456−6−5−4−3−2−11234560ABC?
A(−3, −1), B(4, −1) and C(4, 3) are three vertices of a rectangle. D shares x with A (−3) and y with C (3), so D = (−3, 3). AB = 3 + 4 = 7 and BC = 1 + 3 = 4. Perimeter = 2 × (7 + 4) = 22, area = 7 × 4 = 28.
P(1, 1), Q(5, 1) and R(5, 5) are three vertices of a square. PQ = 4, so all sides are 4. S shares x with P and y with R: S = (1, 5).
E(−2, −2), F(3, −2) and G(3, 4) form a right triangle with the right angle at F, because EF is horizontal and FG is vertical. EF = 2 + 3 = 5 units and FG = 2 + 4 = 6 units.
Class activity

Shape detective. Choose three points that make three corners of a rectangle on squared paper. Hide the fourth. Give the three points to a friend and ask for the fourth.

Practice
1
K(−4, −3), L(2, −3) and M(2, 1) are vertices of a rectangle. Find the fourth vertex N.
2
Find the perimeter and the area of rectangle KLMN.
3
W(−1, 2) and X(3, 2) are two vertices of a square. Give one pair of possible other vertices.
4
The points (0, 0), (4, 0) and (0, −3) are joined. Which angle is a right angle, and what are the lengths of the two shorter sides?
5
A rectangle has vertices (−2, 1), (3, 1), (3, −3) and (−2, −3). Find its sides.