Geometry and measures, Year 10: similar triangles, constructions and loci

Sine (sin)

HintThe first three letters of SOH CAH TOA spell it out.

The 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.

adjacent

HintIt is the shorter side that touches the angle θ.

The 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.

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.

The 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.

10 cm

HintBC is across the triangle from angle A, so pick the ratio that links that side with the longest one.

The 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.

8

HintThe side you want sits underneath the fraction this time, so a single multiplication will not do.

The whyQR is opposite angle P and PQ is adjacent to it, so tan 37° = 6 ÷ PQ. Rearranging, PQ = 6 ÷ 0.75 = 8 cm.

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?

The 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.

An equilateral triangle

HintAll three distances between the points were set by one unchanged compass width.

The 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°.

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.

The 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°.

They are the same distance from P

HintThe compass width is not altered between making one mark and the other.

The 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.

A locus

HintA five-letter word from Latin, whose plural ends in -i.

The 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.

A circle, centre A, radius 3 cm

HintOpen a pair of compasses to that width and put the point on A.

The 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.

The perpendicular bisector of AB

HintOne such place is halfway between them; the rest lie on a straight line through it.

The 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.

bisector

HintIt splits the angle into two equal parts.

The 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 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.

The 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.