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

Sine (sin)

提示The first three letters of SOH CAH TOA spell it out.

为什么For 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

提示It is the shorter side that touches the angle θ.

为什么The 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

提示The two triangles have equal angles, so think about what an enlargement does to a ratio of two sides.

为什么The 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

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

为什么BC 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

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

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

perpendicular

提示Picture walking from a spot in a field to a straight road by the quickest route: at what angle do you meet the road?

为什么Any 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

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

为什么Each 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

提示P is equally far from A and from B, so it lies on a line you already know how to construct.

为什么Every 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

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

为什么With 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

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

为什么For 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

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

为什么Every 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

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

为什么Every 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

提示It splits the angle into two equal parts.

为什么A 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

提示Near an end, the closest part of the segment is the endpoint itself, so think about the locus round a single point.

为什么Beside 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.