ζYear 10 – End of Year RevisionDr Frost

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Non-Right Angled Triangles

Using formula for area of non-right angled triangle:

A = ½ × 2x × x × sin(30) = ½ x2

So 2A = x2

So x = √(2A)

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Non-Right Angled Triangles

Angle CAB = 67°Using sine rule:

(CB / sin67) = (8.7 / sin64)So CB = 8.91015cm

Area = ½ x 8.7 x 8.91015 x sin 49 = 29.3cm2

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Trigonometry

Length BD = √(162 – 122) = √112

sin(40) = √112 / CDCD = √112 / sin(40) = 16.46cm

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Circle Theorems

a) 55°. ABO is a right angle because line AC is a tangent to the circle.

b) 55°. Reason 1: By the alternate segment theorem, angle between chord and tangent is same as angle in alternate segment. Reason 2: Since BE is the diameter, BDE is 90°. And angles in triangle BDE add up to 180°.

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Circle Theorems

a)54

b)27Because angle at circumference is half angle at centre.

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Circle Theorems

Angles of quadrilateral ADOB add up to 360, but angles ABO and ABO are 90 (as AD and AB are tangents of the circle) so DOB = 130.

BCD = 65 as angle at circumference is half angle at centre.

(Alternatively, if you added a line from D to B, you could have used the alternate segment theorem)

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Angles

Exterior angle of hexagon is 360/6 = 60, so interior angle is 180 – 60 = 120.Exterior angle of octagon is 360/8 = 45, so interior angle is 180 – 45 = 135.

Thereforex = 360 – 120 – 135 = 105.

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Angles

Since Tile B is regular, it’s an equilateral triangle with angles 60. Interior angle of Tile A must be (360 – 6)/2 = 150 by using angles around this point.

Therefore exterior angle is 30.Then if Tile A has n sides, 360/n = 30, so n = 12.

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Converting between units

15m2 = 150,000 cm2

23km3= 23,000,000,000 m3

230mm2 = 2.3 cm2

434mm3 = 0.434 cm3

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Congruence and Similarity

a)If congruent, we need to prove either all the sides are the same, or a mixture of sides and angles (i.e. SSS, SAA, SSA or AAA)

Since ABC is equilateral, AB = AC. Both triangles share the same side AD. Angle ADB = ADC. Therefore triangles ABD and ACD are congruent.

b) BD = DC (since congruent triangles)AB = BC (since equilateral triangle)BC = BD + DC = 2BDTherefore BD = ½ AB

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Congruence and Similarity

a) ED / 8 = 6 / 4So ED = 12cm

b) The ratio of lengths AB to CE is 2:3. So the ratio of length AC : length CD is 2:3.If the total length is 25cm, then AC = (2/5) x 25 = 10cm

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Proportionality

M = kL3

160 = k x 23

So k = 160 / 8 = 20

When L = 3, M = 20 x 33 = 540

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x 16 8 24y 10 5 15?

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Given that y is proportional to x, find the missing values.

Simultaneous Equations

x = 3, y = -2?

Area and Perimeter of Shapes

Perimeter:x – 1 + 3x + 3x + 1 = 56So 7x = 56, therefore x = 8

Area is therefore:½ x 24 x 7 = 84.

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Enlargement

Method 1: Draw enlarged triangle and find its area.Points are (1,1), (3,1) and (2.5, 2.5). Therefore area is ½ x 2 x 1.5 = 1.5.

Method 2: Area of original triangle is ½ x 4 x 3 = 6.If length is enlarged by scale factor of ½, then area enlarged by scale factor of (½)2 = ¼ . So area is 6 x ¼ = 1.5.

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Surdsa)5√2 / 2

b) 4 + 2√3 + 2√3 + 3 – (4 – 2√3 – 2√3 + 3)Notice how I’ve used brackets for the second expanded expression, so that I negate the terms properly...= 4 + 4√3 + 3 – 4 + 4√ 3 – 3= 8√3

Alternatively, we could have noticed we have the difference of two squares:(2 + √3 + 2 – √3)(2 + √3 – 2 + √3)= 4(2√3) = 8√3

= 4√2 – 3So a = 4 and b = -3

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Surds

= 6 + 2√8 + 3√2 + √16= 6 + 2√4√2 + 3√2 + 4= 6 + 4√2 + 3√2 + 4= 10 + 7√2

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Coordinate Geometry

On line AD, when x = 0, y = 6 (giving point D). When y = 0, x = 3 (giving point A).

Find the perpendicular line to y = -2x + 6 at the point A. We get y = 0.5x – 1.5.When x = 0, y = -1.5 (giving point P).

Therefore length PD = 1.5 + 6 = 7.5

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Factorisation and Simplification= (x+7)(x-7)

= 3x4y3/2

= (2t + 1)(t + 2)

In the above factorisation, we have the product of two numbers which must both be at least 1.

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x = 4 √(16 – (4 x 3 x -2))

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4 √406 = = 1.72 or -0.38

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Factorisation and Simplification

(2x + y)(x + y)(x + y)(x - y)

2x + yx – y

= =

5(2x+1)2 = (5x-1)(4x+5)5(4x2 + 4x + 1) = 20x2 + 21x – 520x2 + 20x + 5 = 20x2 + 21x – 520x + 5 = 21x – 510 = x

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Multiply everything by x first:x2 + 3 = 7xSo x2 – 7x + 3 = 0This won’t factorise, so use quadratic formula. But make sure you leave your answer in surd form and not decimal form, otherwise your answer won’t be be ‘exact’!

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a) p9

b) q3

c) 2ud) 3wy3

Simpliy (m-2)5 Answer: m-10 or 1 / m10?

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i) 1ii) 8iii) = (8/27)2/3 = (2/3)2 = 4/9

i.e. ‘Flip (if negative power), then Root (if fractional power), then Power’.

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Factorisation and Simplification

Probability

a) 2/7 x 1/6 = 2/42 = 1/21b) Possibilities for tiles are 1-2, 1-3 and 2-3.

Probabilities for each are 2/7 x 3/6 = 6/42, 2/7 x 2/6 = 4/42 and 3/7 x 2/6 = 6/42. So total probability is 16/42 = 8/21.

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Sequences

a) x2 + 2xy + y2

b) 3n – 2

c) Remember that a term is in the sequence if there is some integer k such that 3k – 2 equals our term.

Square of a term in the sequence:(3n – 2)2 = 9n2 – 6n + 4 = 9n2 – 6n + 6 – 2 = 3(n2 – 3n + 2) – 2Therefore it must be a term in the sequence, more specifically, the (n2 – 3n + 2)th term.

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Trial and Improvement

x = 1.5 : 1.53 + (10 x 1.5) = 18.375 Too smallx = 1.8 : 1.83 + (10 x 1.8) = 23.832 Still too smallx = 1.9 : 1.93 + (10 x 1.9) = 25.859 Too bigx = 1.85 : 1.853 + (10 x 1.85) = 24.832 Too small

Therefore solution correct to 1dp must be 1.9.It’s important that once you’ve identified the two closest solutions either side (1.8 and 1.9) you try the midpoint (1.85). Otherwise you won’t know which of the two is the correct solution.

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Inequalities

Answer: -1, 0, 1, 2, 3

2x > x - 6

-x + 1 ≤ 6

x > - 6

x ≥ -5

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1 ≤ 2x + 3 < 9 -1 ≤ x < 3? ?

Solve the following: