Every card in Geometry and measures, Year 10: similar triangles, constructions and loci
The whole deck, in order — so you can read it through before your child ever sees it.
← Back to Geometry and measures, Year 10: similar triangles, constructions and loci
- The trigonometric ratio equal to opposite ÷ hypotenuse
Sine (sin)
HintThe first three letters of SOH CAH TOA spell it out.
WhyFor an angle in a right-angled triangle, the opposite side is the one across from the angle and the hypotenuse is the longest side. Their ratio is the sine of the angle. AQA expects the three ratios to be known; they are not given in the exam.
- In a right-angled triangle, cos θ = ____ ÷ hypotenuse.
adjacent
HintIt is the shorter side that touches the angle θ.
WhyThe adjacent side lies next to the angle and is not the hypotenuse. sin uses opposite ÷ hypotenuse, cos uses adjacent ÷ hypotenuse and tan uses opposite ÷ adjacent.
- Two right-angled triangles each contain an angle of 35°. Every side of one triangle is three times as long as the matching side of the other. How does the value of opposite ÷ hypotenuse for the 35° angle compare in the two triangles?
It is the same in both
HintThe two triangles have equal angles, so think about what an enlargement does to a ratio of two sides.
WhyThe triangles have the same three angles, so they are similar. Enlarging multiplies the opposite side and the hypotenuse by the same scale factor, which cancels when one is divided by the other. That fixed value is what sin 35° means.
- Triangle ABC has a right angle at B. Angle A is 30° and the hypotenuse AC is 20 cm. Using sin 30° = 0.5, how long is side BC?
10 cm
HintBC is across the triangle from angle A, so pick the ratio that links that side with the longest one.
WhyBC is opposite angle A and AC is the hypotenuse, so sin 30° = BC ÷ 20. Multiplying both sides by 20 gives BC = 20 × 0.5 = 10 cm.
- Triangle PQR has a right angle at Q. Angle P is 37° and side QR is 6 cm. Using tan 37° = 0.75, how long is side PQ, in cm? (number only)
8
HintThe side you want sits underneath the fraction this time, so a single multiplication will not do.
WhyQR is opposite angle P and PQ is adjacent to it, so tan 37° = 6 ÷ PQ. Rearranging, PQ = 6 ÷ 0.75 = 8 cm.
- The shortest distance from a point to a straight line is the ____ distance.
perpendicular
HintPicture walking from a spot in a field to a straight road by the quickest route: at what angle do you meet the road?
WhyAny other straight path from the point to the line is the hypotenuse of a right-angled triangle whose shorter side is the perpendicular, so it must be longer. "The distance from a point to a line" always means this one.
- To construct an angle of 60° with compasses, you draw an arc centred on one end of a line. Keeping the compasses at the same width, you draw a second arc centred on the point where the first arc crosses the line. Joining the first centre to the point where the two arcs cross gives the 60° angle. Which shape do the two centres and that crossing point make?
An equilateral triangle
HintAll three distances between the points were set by one unchanged compass width.
WhyEach side of the triangle is one compass radius, so all three sides are equal. A triangle with three equal sides has three equal angles, and 180° ÷ 3 = 60°.
- To construct the perpendicular from a point P to a line, you first draw an arc centred on P that crosses the line at two points, A and B. Which standard construction on A and B finishes the job?
The perpendicular bisector of AB
HintP is equally far from A and from B, so it lies on a line you already know how to construct.
WhyEvery point on the perpendicular bisector of AB is the same distance from A and from B. P is such a point, because both are one arc radius away, so the bisector passes through P and meets the line at 90°.
- To construct a perpendicular to a line at a point P that lies ON the line, you first put the compass point on P and mark the line once on each side of P. What must be true of the two marks?
They are the same distance from P
HintThe compass width is not altered between making one mark and the other.
WhyWith the two marks equally far from P, P is the midpoint between them. Constructing the perpendicular bisector of the segment that joins the marks then gives a line through P at 90° to the original line.
- The set of all points that obey a given rule
A locus
HintA five-letter word from Latin, whose plural ends in -i.
WhyFor example, all the points 3 cm from one fixed point make a circle. A locus can be a line, a curve or a whole region.
- On a flat sheet of paper, describe the locus of points that are exactly 3 cm from a fixed point A.
A circle, centre A, radius 3 cm
HintOpen a pair of compasses to that width and put the point on A.
WhyEvery point on the circle is one radius from its centre, and no other point is. Points less than 3 cm from A fill the inside of the circle.
- Describe the locus of points that are the same distance from two fixed points, A and B.
The perpendicular bisector of AB
HintOne such place is halfway between them; the rest lie on a straight line through it.
WhyEvery point on the perpendicular bisector is equally far from A and from B, and every other point is closer to one of them. It is constructed with crossing arcs drawn from A and from B.
- Two straight lines meet at a point. The locus of points inside the angle that are the same distance from both lines is the angle ____.
bisector
HintIt splits the angle into two equal parts.
WhyA point on the angle bisector has equal perpendicular distances to the two arms of the angle. It is constructed with an arc across both arms and then two crossing arcs.
- A line segment AB is 6 cm long. The locus of points exactly 2 cm from the segment is made of two straight lines parallel to AB, joined by a curve at each end. What shape is the curve at each end?
A semicircle of radius 2 cm
HintNear an end, the closest part of the segment is the endpoint itself, so think about the locus round a single point.
WhyBeside the segment the nearest point is straight across, which gives two parallel lines 2 cm away. Beyond an end the nearest point is A or B, and points 2 cm from one point lie on a circle, half of which is needed at each end. The whole locus looks like a running track.
1★ GCSE-MATH-GEO-0001
2★ GCSE-MATH-GEO-0002
3★ GCSE-MATH-GEO-0003
4★ GCSE-MATH-GEO-0004
5★ GCSE-MATH-GEO-0005
6★ GCSE-MATH-GEO-0006
7★ GCSE-MATH-GEO-0007
8★ GCSE-MATH-GEO-0008
9★ GCSE-MATH-GEO-0009
10★ GCSE-MATH-GEO-0010
11★ GCSE-MATH-GEO-0011
12★ GCSE-MATH-GEO-0012
13★ GCSE-MATH-GEO-0013
14★ GCSE-MATH-GEO-0014
Keep what you learn
Here, nothing is saved. In your child’s own sky every card is scheduled — it comes back just before they’d forget it — and the professor who wrote it is one tap away.