Geometry and measures, Year 10: similar shapes, Pythagoras and vectors

k³

HintA volume needs three lengths multiplied together, and each of them has been scaled.

The whyEach length is multiplied by k. An area is a length × a length, so it is multiplied by k × k; a volume is three lengths multiplied, so by k × k × k. With k = 2, areas are 4 times and volumes 8 times as large.

45 cm²

HintThe scale factor has to be applied twice, once for each direction the shape is stretched in.

The whyArea scale factor = 3² = 9. 5 × 9 = 45 cm².

2400 ml

HintThe taller bottle is also wider and deeper by the same factor.

The whyLength scale factor = 20 ÷ 10 = 2. Volume scale factor = 2³ = 8. 300 × 8 = 2400 ml.

20

HintGo from the ratio of the areas back to the ratio of the lengths by undoing a square.

The whyArea scale factor = 100 ÷ 16 = 6.25. Length scale factor = √6.25 = 2.5. 8 × 2.5 = 20 cm.

8 : 27

HintStep down to the ratio of lengths first, then step up to three dimensions.

The whyAreas are in the ratio of the squares of the lengths, so the lengths are in the ratio √4 : √9 = 2 : 3. Volumes are in the ratio of the cubes: 2³ : 3³ = 8 : 27.

sum

HintIt is the result of an addition.

The whyWith hypotenuse c and shorter sides a and b, this is a² + b² = c². AQA expects the formula to be known; it is not given in the exam.

PQ² + QR² = PR²

HintDecide which side lies across from the right angle; it stands alone in the formula.

The whyThe hypotenuse is opposite the right angle. The right angle is at Q, so the hypotenuse is the side that does not touch Q, which is PR. The squares of the other two sides add up to its square.

Top 5, bottom −2

HintAcross goes above, up goes below, and a move the "wrong" way takes a minus sign.

The whyThe top number of a column vector is the movement across (right is positive) and the bottom number is the movement up (down is negative). So 5 right and 2 down is written with 5 above −2 inside one pair of brackets.

left

HintThe top number works like the x-direction on a graph.

The whyThe top number is the horizontal movement, positive to the right and negative to the left. The bottom number is the vertical movement, positive up and negative down.

(−2, 9)

HintAdd the top number to the x-coordinate and the other number to the y-coordinate.

The whyx: 2 + (−4) = −2. y: 3 + 6 = 9. A translation adds the same vector to every point of a shape, so the shape keeps its size and its orientation.

3

HintCompare the two x-coordinates: finish take away start.

The whyTop number = change in x = 4 − 1 = 3. The bottom number is the change in y: 1 − 5 = −4. So the vector has 3 on top and −4 underneath.

Top −3, bottom 2

HintTo undo a slide, go the same distances in the opposite directions.

The whyReversing a translation changes the sign of both numbers: 3 right and 2 down is undone by 3 left and 2 up.