Geometry and measures, Year 8: areas, units, bearings and bisectors

40 cm²

HintOne of the three numbers is not needed.

The whyArea = base × perpendicular height = 8 × 5. The slant side plays no part in the area.

9 cm

HintUndo the multiplication.

The whyBase × 4 = 36, so the base is 36 ÷ 4 = 9 cm. The answer is a length, so the unit is cm, not cm².

Cutting off a triangle and sliding it across makes a rectangle

HintImagine scissors and one straight snip from a top corner.

The whyCut a right-angled triangle from one end and move it to the other. The result is a rectangle with the same base and the same height, and no area has been lost.

It gets smaller

HintIts base is unchanged, so think about how tall it now stands.

The whyArea is base × perpendicular height. The base and the slant sides are unchanged, but the shape is now lower, so the area shrinks.

10,000 cm²

HintA metre-square tile is 100 cm along each edge.

The why1 m² is a square 100 cm by 100 cm, and 100 × 100 = 10,000. The length factor of 100 is squared.

30,000 cm²

HintThe scale factor for length gets squared.

The whyEach square metre holds 10,000 cm², so 3 × 10,000 = 30,000 cm².

2,000,000 cm³

HintA metre cube is 100 cm in three directions.

The why1 m³ = 100 × 100 × 100 = 1,000,000 cm³, so 2 m³ is two million. For volume, the length factor is cubed.

0.5 m²

HintGoing to the bigger unit means dividing.

The why5,000 ÷ 10,000 = 0.5. It takes ten thousand square centimetres to fill one square metre, so 5,000 fills half of one.

Clockwise

HintStart facing north and turn towards east.

The whyA bearing is the angle turned from north, in that direction, to face where you are heading. It is written with three figures.

090°

HintA quarter turn from north, written with three figures.

The whyEast is 90° round from north. Bearings always have three figures, so a zero is put in front.

250°

HintComing back means a half turn.

The whyThe return direction is exactly opposite, so add 180°: 70 + 180 = 250.

020°

HintHalf a turn back, and keep three figures.

The whyAdding 180° would go past 360°, so subtract: 200 − 180 = 20, written as 020°.

The perpendicular bisector

HintIts two words mean 'at 90°' and 'cuts in two'.

The whyIt is constructed by drawing arcs of equal radius from each end of the segment and joining the two points where the arcs cross.

It is the same distance from A and B

HintThink about where you could stand to be fair to both ends.

The whyThat is why the construction works: the arcs have equal radii, so the points where they cross are equally far from A and from B.

The two arms of the angle

HintA ball rolling along it would stay midway between two walls.

The whyThe bisector cuts the angle into two equal angles, so it runs exactly midway between the two lines that form it.

No — leave them as evidence of the method

HintThey show how the line was found.

The whyConstruction arcs prove that the line was constructed and not measured or guessed. Without them the method cannot be seen.