Geometry and measures, Year 10: proof, enlargement and circle theorems

SSS

HintFind the pairs of equal lengths: one is given, one comes from the midpoint, and one line belongs to both triangles.

The whyAB = AC (given), BM = CM (M is the midpoint) and AM is common to both triangles. Three pairs of equal sides is SSS. Because the triangles are congruent, angle B = angle C, which proves that the base angles of an isosceles triangle are equal.

alternate

HintAB and DC are parallel, and the diagonal crosses both of them.

The whyAB is parallel to DC and AC cuts across both, so angles BAC and DCA are alternate angles and are equal. The same argument gives angle BCA = angle DAC. With AC common, triangles ABC and CDA are congruent (ASA), which proves that opposite sides of a parallelogram are equal.

x + 2x + 3x = 180, so x = 30 and the largest angle is 3x = 90°

HintUse the fact about the total of a triangle's angles to form an equation.

The whyAngles in a triangle add up to 180°, so 6x = 180 and x = 30. The angles are 30°, 60° and 90°. A proof sets out each step with its reason and ends with the statement that was asked for.

It only checks some triangles, not all of them

HintThink about how many different shapes of this kind exist, compared with how many were tested.

The whyExamples can suggest a rule (a conjecture) but cannot rule out an exception that has not been tried, and measurements are never exact. A proof uses reasoning, such as congruent triangles, that applies to every case at once.

Their three pairs of angles are equal

HintOne corner belongs to both triangles, and the parallel lines do the rest at D and at E.

The whyAngle A is common. Angle ADE = angle ABC and angle AED = angle ACB, because they are corresponding angles on the parallel lines DE and BC. Equal angles make the triangles similar, so their sides are in the same ratio.

10 cm

HintCompare the two shortest sides to find the multiplier, which this time is less than 1.

The whyThe shortest sides match, so the scale factor from larger to smaller is 6 ÷ 9 = 2/3. The longest side is then 15 × 2/3 = 10 cm. A scale factor between 0 and 1 makes the shape smaller.

smaller

HintMultiply any length by a fraction such as 1/2 and see what happens to it.

The whyEvery length is multiplied by the scale factor. With a scale factor of 1/2, a 10 cm side becomes 5 cm. The transformation is still called an enlargement even though the shape shrinks.

(2, 4)

HintWith this centre, each coordinate is simply multiplied by the scale factor.

The whyWhen the centre is the origin, the distances from the centre across and up are the coordinates themselves, so both are multiplied by 1/2: (4 × 1/2, 8 × 1/2) = (2, 4).

(5, 3)

HintFind how far A is from the centre, across and up, and shrink that journey, not the coordinates.

The whyFrom the centre (3, 1) to A is 6 across and 6 up. One third of that is 2 across and 2 up. Starting again from the centre: (3 + 2, 1 + 2) = (5, 3).

3/4

HintDivide the new length by the old one, in that order.

The whyScale factor = image length ÷ original length = 9 ÷ 12 = 3/4. It is less than 1 because the image is smaller than the original.

55

HintThe angle at the middle of the circle and the angle at the edge stand on the same arc; one is double the other.

The whyThe angle at the centre is twice the angle at the circumference standing on the same arc, so angle ACB = 110 ÷ 2 = 55°.

55°

HintA triangle drawn on a diameter with its third corner on the circle always contains one particular angle.

The whyThe angle in a semicircle is 90°, so angle ACB = 90°. The angles of triangle ABC add up to 180°, so angle CBA = 180 − 90 − 35 = 55°.

40° — angles in the same segment are equal

HintThe two corners at C and D both look at chord AB from one side of it.

The whyAngles at the circumference that stand on the same arc, in the same segment, are equal. Each is half the angle that AB makes at the centre, which is why they match.

A cyclic quadrilateral

HintThe word comes from the same root as "cycle" and "circle".

The whyIts opposite angles add up to 180°. A four-sided shape with only three vertices on the circle, or with the centre as a vertex, is not cyclic, so that rule does not apply to it.

105

HintB and D face each other across the shape, and such a pair of corners has a fixed total.

The whyOpposite angles of a cyclic quadrilateral add up to 180°, so angle ADC = 180 − 75 = 105°.

Two of its sides are radii, so they are equal

HintAsk what OA, OB and OC all are in this circle.

The whyOA = OC and OB = OC because every radius of a circle has the same length. Isosceles triangles have equal base angles, and the exterior angle of each triangle is the sum of those two equal angles, which gives the doubling.