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

k³

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

为什么Each 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

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

为什么Area scale factor = 3² = 9. 5 × 9 = 45 cm².

2400 ml

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

为什么Length scale factor = 20 ÷ 10 = 2. Volume scale factor = 2³ = 8. 300 × 8 = 2400 ml.

20

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

为什么Area scale factor = 100 ÷ 16 = 6.25. Length scale factor = √6.25 = 2.5. 8 × 2.5 = 20 cm.

8 : 27

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

为什么Areas 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

提示It is the result of an addition.

为什么With 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

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

为什么The 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

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

为什么The 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

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

为什么The 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)

提示Add the top number to the x-coordinate and the other number to the y-coordinate.

为什么x: 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

提示Compare the two x-coordinates: finish take away start.

为什么Top 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

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

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