Geometry and the Coordinate Plane: math lesson, Grade 5

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Math lessons Grade 5 : Geometry and the Coordinate Plane — Zyro the alien explorer of Planète Maths

Maps, video games and city streets all use the same idea: a grid that gives every place an address. In this chapter you will learn to find and graph points on the coordinate plane, and you will also sort two-dimensional figures like a real geometer, by looking at their sides, angles and parallel lines.

Sommaire

1. The coordinate plane

The coordinate plane is made of two number lines that cross at a right angle. The horizontal number line is the x-axis. The vertical number line is the y-axis. They meet at the point called the origin.

Definition: origin and ordered pairThe origin is the point \((0, 0)\) where the two axes cross. Every point is named by an ordered pair \((x, y)\). The first number, \(x\), tells how far to move right. The second number, \(y\), tells how far to move up.

Look at the figure. To reach point A you start at the origin, move 4 units right and then 3 units up. So A is at \((4, 3)\). In this chapter we work in the first quadrant, the part of the plane where both numbers are 0 or greater.

2. Plotting ordered pairs

Method: plot a point

Start at the origin \((0, 0)\).
Move right along the x-axis by the first number.
From there, move up by the second number.
Draw a dot and label it with its letter.

Example 1Plot \((6, 2)\) and \((2, 6)\). For \((6, 2)\) you go 6 units right and 2 units up. For \((2, 6)\) you go 2 units right and 6 units up. These are two different points, because the order of the numbers matters.

Watch outDo not swap the numbers. The pair \((x, y)\) always lists the horizontal move first. A point on the x-axis has \(y = 0\), like \((5, 0)\). A point on the y-axis has \(x = 0\), like \((0, 5)\).

3. Problems on the coordinate plane

A coordinate plane can be a map. Each grid unit stands for a real distance, such as 1 block, 100 meters or half a mile. When two points share the same first number, they sit on a vertical line. When they share the same second number, they sit on a horizontal line. Then the distance is easy: subtract the numbers that are different.

Example 2: a park mapOn the map, the library is at \((2, 5)\), the pool at \((6, 5)\) and the cafe at \((6, 1)\). Each unit is 50 yards. From the library to the pool, only the first number changes: \(6 - 2 = 4\) units, so the walk is \(4 \times 50 = 200\) yards. From the pool to the cafe, only the second number changes: \(5 - 1 = 4\) units, which is also 200 yards. The whole trip is \(200 + 200 = 400\) yards (about 366 meters).

4. Parallel and perpendicular lines

Definition: parallel and perpendicularTwo lines are parallel if they stay the same distance apart and never meet. Two lines are perpendicular if they meet and form a right angle (a square corner of 90°).

On the coordinate plane, the x-axis and the y-axis are perpendicular. Any two horizontal lines are parallel to each other, and any horizontal line is perpendicular to any vertical line. The sides of a door frame, the rails of a train track and the edges of a notebook page show both ideas.

5. Attributes of quadrilaterals

A quadrilateral is a closed figure with 4 straight sides. We sort quadrilaterals by their attributes: parallel sides, equal sides and right angles.

Shape	Parallel sides	Equal sides	Right angles
Trapezoid	at least 1 pair	not required	not required
Parallelogram	2 pairs	opposite sides equal	not required
Rectangle	2 pairs	opposite sides equal	4
Rhombus	2 pairs	all 4 sides equal	not required
Square	2 pairs	all 4 sides equal	4

Example 3A quadrilateral has 4 sides of length 5 in, and its opposite sides are parallel, but its corners are not square. It has 4 equal sides, so it is a rhombus. Its perimeter is \(4 \times 5 = 20\) in, which is 50.8 cm.

6. Classifying triangles

A triangle can be named in two ways. By its sides: equilateral (3 equal sides), isosceles (at least 2 equal sides) or scalene (no equal sides). By its angles: acute (3 angles smaller than 90°), right (one angle of exactly 90°) or obtuse (one angle larger than 90°).

Example 4A triangle has sides of 6 cm, 6 cm and 9 cm and one angle of about 97°. Two sides are equal, so it is isosceles. One angle is larger than 90°, so it is also obtuse. We call it an obtuse isosceles triangle.

Zyro’s tipOn my home planet we say “iso” means “same”. If you see the little tick marks on two sides, the triangle is isosceles. Check the angles separately: the sides and the angles give two different names!

7. The hierarchy of two-dimensional figures

Some shapes belong to a bigger family. A shape in a lower box has all the attributes of the boxes above it, plus something extra.

Property: the family treeEvery square is a rectangle and a rhombus. Every rectangle and every rhombus is a parallelogram. Every parallelogram is a trapezoid and a quadrilateral. The reverse is not true: a rectangle with unequal sides is not a rhombus.

This is why a square can have many names. The most specific name, square, tells you the most about it.

8. Venn diagrams for shapes

A Venn diagram uses overlapping circles to sort objects. Each circle holds the shapes with one attribute. The overlap holds the shapes that have both attributes.

Example 5In the diagram, the left circle is “4 equal sides” and the right circle is “4 right angles”. A rhombus with slanted corners has 4 equal sides only, so it goes in the left part. A rectangle with a long side and a short side goes in the right part. A square has both attributes, so it goes in the overlap.

Key takeaways

The origin is \((0, 0)\). In an ordered pair \((x, y)\), move right \(x\) units first, then up \(y\) units.
Points that share the same first number are on a vertical line. Points that share the same second number are on a horizontal line.
Parallel lines never meet. Perpendicular lines meet at 90°.
A parallelogram has 2 pairs of parallel sides. A rectangle has 4 right angles. A rhombus has 4 equal sides. A square has both.
Triangles are named by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).
Every square is a rectangle, a rhombus, a parallelogram and a quadrilateral.
In a Venn diagram, the overlap holds the shapes with both attributes.