Every card in Geometry and measures, Year 10: similar shapes, Pythagoras and vectors
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- Two solids are similar, and every length in the larger one is k times the matching length in the smaller. The volume of the larger solid is then ____ times the volume of the smaller.
k³
HintA volume needs three lengths multiplied together, and each of them has been scaled.
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.
- Two similar shapes have a length scale factor of 3. The smaller has an area of 5 cm². What is the area of the larger?
45 cm²
HintThe scale factor has to be applied twice, once for each direction the shape is stretched in.
WhyArea scale factor = 3² = 9. 5 × 9 = 45 cm².
- Two bottles are mathematically similar. Their heights are 10 cm and 20 cm. The smaller bottle holds 300 ml. How much does the larger bottle hold?
2400 ml
HintThe taller bottle is also wider and deeper by the same factor.
WhyLength scale factor = 20 ÷ 10 = 2. Volume scale factor = 2³ = 8. 300 × 8 = 2400 ml.
- Two similar shapes have areas of 16 cm² and 100 cm². A side of the smaller shape is 8 cm. How long is the matching side of the larger shape, in cm? (number only)
20
HintGo from the ratio of the areas back to the ratio of the lengths by undoing a square.
WhyArea scale factor = 100 ÷ 16 = 6.25. Length scale factor = √6.25 = 2.5. 8 × 2.5 = 20 cm.
- Two similar solids have surface areas in the ratio 4 : 9. What is the ratio of their volumes?
8 : 27
HintStep down to the ratio of lengths first, then step up to three dimensions.
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.
- Pythagoras' theorem: in a right-angled triangle, the square of the hypotenuse equals the ____ of the squares of the other two sides.
sum
HintIt is the result of an addition.
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.
- Triangle PQR has a right angle at Q. Write Pythagoras' theorem for this triangle, using the side names PQ, QR and PR.
PQ² + QR² = PR²
HintDecide which side lies across from the right angle; it stands alone in the formula.
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.
- A translation moves every point 5 to the right and 2 down. Write it as a column vector, giving the top number and then the bottom number.
Top 5, bottom −2
HintAcross goes above, up goes below, and a move the "wrong" way takes a minus sign.
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.
- In the column vector of a translation, a negative top number means the shape moves to the ____.
left
HintThe top number works like the x-direction on a graph.
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.
- The point (2, 3) is translated by the column vector with −4 on top and 6 underneath. Where does the point land?
(−2, 9)
HintAdd the top number to the x-coordinate and the other number to the y-coordinate.
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.
- A translation maps the point (1, 5) to the point (4, 1). What is the top number of its column vector? (number only)
3
HintCompare the two x-coordinates: finish take away start.
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.
- A translation takes shape A to shape B. Its column vector has 3 on top and −2 underneath. What are the top and bottom numbers of the column vector that takes B back to A?
Top −3, bottom 2
HintTo undo a slide, go the same distances in the opposite directions.
WhyReversing a translation changes the sign of both numbers: 3 right and 2 down is undone by 3 left and 2 up.
1★ GCSE-MATH-GEO-0060
2★ GCSE-MATH-GEO-0061
3★ GCSE-MATH-GEO-0062
4★ GCSE-MATH-GEO-0063
5★ GCSE-MATH-GEO-0064
6★ GCSE-MATH-GEO-0065
7★ GCSE-MATH-GEO-0066
8★ GCSE-MATH-GEO-0067
9★ GCSE-MATH-GEO-0068
10★ GCSE-MATH-GEO-0069
11★ GCSE-MATH-GEO-0070
12★ GCSE-MATH-GEO-0071
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