The sum of the degrees of all the vertices in the graph above is
- `6`
- `9`
- `11`
- `12`
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The sum of the degrees of all the vertices in the graph above is
`D`
`text(Total Degrees)` | `=1+3+2+2+2+2` | |
`=12` |
`rArr D`
In the diagram, `ABCDE` is a regular pentagon. The diagonals `AC` and `BD` intersect at `F`.
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i. |
`text(Sum of all internal angles)`
`= (n-2) xx 180°`
`= (5-2) xx 180°`
`= 540°`
`:. /_ABC= 540/5= 108°`
ii. `BA = BC\ \ text{(equal sides of a regular pentagon)}`
`:. Delta BAC\ text(is isosceles)`
`/_BAC= 1/2 (180-108)=36^{\circ} \ \ \ text{(base angle of}\ Delta BAC text{)}`
A quadrilateral is pictured below.
What is the value of `x`? (3 marks)
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`126^@`
`text{Sum of exterior angles = 360°}`
`y^{\circ}` | `=360-(127+114+65)` | |
`=360-306` | ||
`=54^{\circ}` |
`:.x^{\circ}=180-54 = 126^{\circ}\ \ \text{(180° in straight line)}`
A pentagon is pictured below, where one internal angle is a right angle.
What is the value of `x`? (3 marks)
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`130^@`
`y^{\circ}=180-100 = 80^{\circ}\ \ \text{(180° in straight line)}`
`text{Sum of exterior angles = 360°}`
`z^{\circ}` | `=360-(70+70+90+80)` | |
`=360-310` | ||
`=50^{\circ}` |
`:.x^{\circ}=180-50 = 130^{\circ}\ \ \text{(180° in straight line)}`
A quadrilateral is drawn below.
What is the value of `x`? (3 marks)
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`117^@`
A regular decagon is pictured below.
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i. `36^@`
ii. `144^@`
i. `text{Sum of exterior angles = 360°}`
`text{Since the decagon is regular, all external angles are equal.}`
`:.x^{\circ}= 360/10 = 36^{\circ}`
ii. `text{Method 1: Using exterior angle}`
`text{Internal angle}` | `=180-\text{exterior angle}` |
`=180-36` | |
`=144^{\circ}` |
`text{Method 2: Using Internal angle sum formula}`
`text{Sum of internal angles}` | `=(n-2) xx 180` |
`=(10-2) xx 180` | |
`=1440^{\circ}` |
`:.\ text{Internal angle}\ = 1440/10 = 144^{\circ}`
A regular nonagon is pictured below.
What is the value of `x`? (2 marks)
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`40^@`
`text{Sum of exterior angles = 360°}`
`text{Since the nonagon is regular, all external angles are equal.}`
`:.x^{\circ}= 360/9 = 40^{\circ}`
A regular pentagon is pictured below.
What is the value of `x`? (2 marks)
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`72^@`
`text{Sum of exterior angles = 360°}`
`text{Since the pentagon is regular, all external angles are equal.}`
`:.x^{\circ}= 360/5 = 72^{\circ}`
A quadrilateral is drawn below.
What is the value of `x`? (3 marks)
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`117^@`
`y^{\circ}=180-75 = 105^{\circ}\ \ \text{(180° in straight line)}`
`text{Sum of exterior angles = 360°}`
`z^{\circ}` | `=360-(130+105+62)` | |
`=360-297` | ||
`=63^{\circ}` |
`:.x^{\circ}=180-63 = 117^{\circ}\ \ \text{(180° in straight line)}`
A pentagon is drawn below.
What is the value of `x`? (3 marks)
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`99^@`
`z^{\circ}=180-110 = 70^{\circ}\ \ \text{(180° in straight line)}`
`text{Sum of exterior angles = 360°}`
`y^{\circ}` | `=360-(72+82+70+55)` | |
`=360-279` | ||
`=81^{\circ}` |
`:.x^{\circ}=180-81 = 99^{\circ}\ \ \text{(180° in straight line)}`
A quadrilateral is drawn below.
What is the value of `x`? (2 marks)
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`103^@`
`text{Sum of exterior angles = 360°}`
`x` | `=360-(105+95+57)` | |
`=360-257` | ||
`=103^{\circ}` |
A five sided polygon is drawn below.
What is the value of `x`? (2 marks)
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`60^@`
`text{Sum of exterior angles = 360°}`
`x` | `=360-(65+85+80+70)` | |
`=360-300` | ||
`=60^{\circ}` |
Igor was designing a shield using 10 congruent (isosceles) triangles, as shown in the diagram below.
How many degrees in the angle marked `x`? (3 marks)
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`72^@`
`text(Angles at centre of circle)\ = 360/10 = 36^@`
`text{Since triangles are isosceles:}`
`180` | `= 36 + 2x` |
`2x` | `= 180-36` |
`= 144 ` | |
`:. x` | `= 72^@` |
`AB` is the diameter of a circle, centre `O`.
There are 3 triangles drawn in the lower semi-circle and the angles at the centre are all equal to `x^@`.
The three triangles are best described as:
`D`
`3x=180^{\circ}\ \=>\ x=60^{\circ} `
`AO=OC=OD=OB\ \ text{(radii of circle)}`
`=>\ text{Since angles opposite equal sides of a triangle are}`
`text(equal, all triangle angles can be found to equal 60°.)`
`:.\ text(The three triangles are equilateral.)`
`=>D`
A regular pentagon, a square and an equilateral triangle meet at a point.
What is the size of the angle `x°`? (3 marks)
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`102°`
`text(Sum of internal angles of pentagon)`
`= (n-2) xx 180`
`= 3 xx 180`
`= 540^@`
`text(Internal angle in pentagon)`
`= 540/5`
`= 108^@`
`:. x` | `= 360-(108 + 90 + 60)` |
`= 102^@` |
A six sided figure is drawn below.
What is the sum of the six interior angles? (2 marks)
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`720^@`
`\text{Method 1}`
`text(Reflex angle) = 360-90 = 270^@`
`:.\ text(Sum of interior angles)`
`= (270 xx 2) + (30 xx 2) + (60 xx 2)`
`= 720^@`
`\text{Method 2}`
`text{Sum of interior angles (formula)}`
`= (n-2) xx 180`
`=4 xx 180`
`= 720^@`
Two identical quadrilaterals fit together to make this regular pentagon.
What is the value of `x`? (2 marks)
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`108^@`
`text(Consider regular pentagon:)`
`text{Sum of internal angles (formula)}`
`= (n-2) xx 180`
`= 3 xx 180`
`= 540^@`
`:. x` | `= 540/5` |
`= 108^@` |
A star is drawn on the inside of a regular pentagon, as shown below.
What is the size of the angle marked `x`? (3 marks)
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`36^@`
`text(Consider the triangle)\ ABC\ \ text(in the pentagon:)`
`text(Total degrees in a pentagon)`
`= 3 xx 180`
`= 540^@`
`text{Internal angle}\ =540/5 = 108^@\ \ \text{(regular pentagon)}`
`DeltaABC\ text(is isosceles)`
`:. x + x + 108` | `= 180` |
`2x` | `= 72` |
`x` | `= 36^@` |
The sum of the interior angles of a 6 sided polygon can be found by first dividing it into triangles from one vertex.
What is the sum of the interior angles of this polygon? (2 marks)
`720\ text(degrees)`
`text{Since the polygon can be divided into 4 separate triangles:}`
`text(Sum of interior angles)`
`= 4 xx 180`
`= 720\ text(degrees)`
`ABCD` is a rhombus. `AC` is the same length as the rhombus sides.
What is the size of `/_ DCB?` (2 marks)
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`120^@`
`Delta ADC and Delta ABC\ text{are equiangular triangles.}`
`/_ DCA = /_ ACB = 60°`
`:. /_ DCB` | `=60 xx 2` |
`=120°` |
Which statement is always true?
`D`
`text{Consider each option:}`
`A:\ \text{Isosceles (not scalene) have two equal angles.}`
`B:\ \text{Only opposite angles in a parallelogram are equal.}`
`C:\ \text{At least one pair of opposite sides of a trapezium are not equal.}`
`D:\ \text{Rhombuses have perpendicular diagonals.}`
`=>D`
Which of these are always equal in length?
`C`
`PQRS` is a parallelogram.
Which of these must be a property of `PQRS`?
`D`
`text{By elimination:}`
`A\ \text{and}\ B\ \text{clearly incorrect.}`
`C\ \text{true if all sides are equal (rhombus) but not true for all parallelograms.}`
`text(Line)\ PS\ text(must be parallel to line)\ QR.`
`=>D`
The sum of the internal angles of a polygon can be calculated by drawing triangles from any given vertex as shown below.
What is the size of the angle marked `x` in the diagram below? (2 marks)
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`107°`
`text{Since the quadrilateral was divided into two triangles}`
`=>\ \text{Sum of internal angles}\ = 2 xx 180 = 360^{\circ}`
`:. x` | `= 360-(103 + 88 + 62)` |
`= 360-253` | |
`= 107°` |
A closed shape has two pairs of equal adjacent sides.
What is the shape?
`C`
`text(Kite.)`
`text{(Note that a rectangle has a pair of equal opposite sides)}`
`=>C`
Which two of the triangles below are congruent? (2 marks)
\(\text{Triangle B and Triangle C}\)
\(\text{Unknown side}\ (x)\ \text{in Triangle B (Pythagoras):}\)
\(x=\sqrt{11^2-(\sqrt{96})^2} = \sqrt{25} = 5\)
\(\Rightarrow\ \text{Triangle B and Triangle C are congruent (SSS)} \)
\(\text{Triangle A can be shown to have different dimensions but this is not necessary.}\)
Which two of the triangles below are congruent? (2 marks)
\(\text{Triangle A and Triangle C}\)
\(\text{Unknown side}\ (x)\ \text{in Triangle A (Pythagoras):}\)
\(x=\sqrt{8^2-(\sqrt{48})^2} = \sqrt{16} = 4\)
\(\Rightarrow\ \text{Triangle A and Triangle C are congruent (SSS)} \)
\(\text{Triangle B can be shown to have different dimensions but this is not necessary.}\)
The two triangles below are congruent.
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i. `text{Congruency test}\ => \ text{SAS} `
ii. `text{Matching corresponding sides:}`
`a=12.1, \ b=7.4`
`A`, `B` and `C` are vertices on the cube below.
What is the best description of `DeltaABC`?
`D`
`AB != AC != BC`
`angleBCA = 90^@`
`DeltaABC\ text(is both right-angled and scalene)`
`:.\ text(Right-angled is the BEST description)`
`=>D`
The area of the rectangle in the diagram below is 15 cm2.
Giving reasons, find the area of the trapezium `ABCD`. (4 marks)
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`30\ text(cm)^2`
`text{Opposite sides of rectangles are equal and parallel}`
`\Delta AEI ≡ \Delta DHJ\ \ text{(RHS)}`
`/_ EAI = /_ BEF\ \ text{(corresponding,}\ AD\ text{||}\ EH )`
`\Delta AEI ≡ \Delta EBF\ \ text{(AAS)}`
`text{Similarly,}\ \Delta DHJ ≡ \Delta HCG\ \ text{(AAS)}`
`=>\ \text{All triangles in diagram are congruent.}`
`text(Rearranging the diagram:)`
`:.\ text(Area of trapezium)`
`= 2 xx 15`
`= 30\ text(cm)^2`
Which one of the following triangles is impossible to draw?
`D`
`text(A right angle = 90°.)`
`text{Since an obtuse angle is greater than 90°, it is impossible for}`
`text(a triangle, with an angle sum less than 180°, to have both.)`
`=>D`
Select the statement that is true about triangle `ABC`.
`C`
`angleA = 180-(60 + 60) = 60^@`
`:. text(All angles are)\ 60^@.`
`:. text(Triangle)\ ABC\ text(is an equilateral.)`
`=>C`
A triangle has two acute angles.
What type of angle couldn't the third angle be?
`D`
`text(A triangle’s angles add up to 180°, and a reflex angle is)`
`text(greater than 180°.)`
`:.\ text(The third angle cannot be reflex.)`
`=>D`
Which of the following triangle types is impossible to draw?
`B`
`text(An equilateral triangle has all angles = 60°.)`
`:.\ text(A right-angled, equilateral triangle is impossible.)`
`=>B`
The diagram shows a circle with centre `O` and radius 5 cm.
The length of the arc `PQ` is 9 cm. Lines drawn perpendicular to `OP` and `OQ` at `P` and `Q` respectively meet at `T`.
Prove that `Delta OPT` is congruent to `Delta OQT`. (2 marks)
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`text(Proof) text{(See Worked Solutions)}`
In the diagram, `AD` is parallel to `BC`, `AC` bisects `/_BAD` and `BD` bisects `/_ABC`. The lines `AC` and `BD` intersect at `P`.
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i. |
`text(Prove)\ /_BAC = /_BCA`
`/_BCA` | `= /_CAD\ \ \ text{(alternate angles,}\ BC\ text(||)\ AD text{)}` |
`/_CAD` | `= /_BAC\ \ \ text{(}AC\ text(bisects)\ /_BAD text{)}` |
`:. /_BAC` | `= /_BCA\ …\ text(as required)` |
ii. `text(Prove)\ Delta ABP ≡ Delta CBP`
`/_BAC` | `= /_BCA\ \ \ text{(from part (i))}` |
`/_ABP` | `= /_CBP\ text{(}BD\ text(bisects)\ /_ABC text{)}` |
`BP\ text(is common)` |
`:. Delta ABP ≡ Delta CBP\ \ text{(AAS)}`
The floor plan of a home unit has been drawn to scale.
What is the total floor area of the home unit in square metres? (2 marks)
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`text{55.5 m}^2`
`text{Conversion: 1000 mm = 1 metre}`
`text{Calculate area by splitting into 2 rectangles}`
`text{Area}` | `= 5.5 × 5 + (4.5 +2.5) xx 4` | |
`= 55.5\ text{m}^2` |
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i. \(\text{Scale factor}\ = \dfrac{\text{Diameter B}}{\text{Diameter A}} = \dfrac{0.8}{2.4} = \dfrac{1}{3} \)
ii. \(\text{Scale factor (B to A)} = \dfrac{\text{Diameter A}}{\text{Diameter B}} = \dfrac{2.4}{0.8} = 3 \)
\(\text{Scale factor (Area)} = 3^2 = 9 \)
\(\therefore\ \text{Area of Circle A = 9 × Area of Circle B} \)
In the figure below, \(ABCD\) is a parallelogram where opposite sides of the quadrilateral are equal.
Prove that a diagonal of the parallelogram produces two triangles that are congruent. (2 marks)
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\(\text{Proof (See Worked Solutions)}\)
\(\text{One of multiple solutions:}\)
\( AB=CD\ \ \text{and}\ \ AC=BD\ \ \text{(given)} \)
\(BC\ \text{is common} \)
\(\therefore\ \Delta ABC \equiv \Delta DCB\ \ \text{(SSS)}\)
In the figure below, \(AB \parallel DE, \ AC = CE\) and the line \(AE\) intersects \(DB\) at \(C\).
Prove that this pair of triangles are congruent. (2 marks)
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\(\text{Proof (See Worked Solutions)}\)
\( \angle ACB = \angle DCE\ \ \text{(vertically opposite)} \)
\(AC = CE\ \ \text{(given)} \)
\(\angle BAC = \angle DEC\ \ \text{(alternate,}\ AB \parallel DE \text{)} \)
\(\therefore\ \Delta ABC \equiv \Delta EDC\ \ \text{(AAS)}\)
In the quadrilateral \(ABCD\), \(AB \parallel CD, \angle BAD = \angle BCD\) and \(\angle DBC = \angle BDA = 90^{\circ} \).
Prove that this pair of triangles are congruent. (2 marks)
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\(\text{Proof (See Worked Solutions)}\)
\(\angle BAD = \angle BCD\ \ \text{(given)} \)
\(\angle DBC = \angle BDA = 90^{\circ} \ \ \text{(given)} \)
\(BD\ \text{is common} \)
\(\therefore\ \Delta BAD \equiv \Delta DCB\ \ \text{(AAS)}\)
In the figure below, \(BE = BC\), \(AB = BD\) and the line \(AD\) intersects \(CE\) at \(B\).
Prove that this pair of triangles are congruent. (2 marks)
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\(\text{Proof (See Worked Solutions)}\)
\( \angle ABE = \angle CBD\ \ \text{(vertically opposite)} \)
\(AB = BD\ \ \text{(given)} \)
\(EB = BC\ \ \text{(given)} \)
\(\therefore\ \Delta ABE \equiv \Delta DBC\ \ \text{(SAS)}\)
In the circle below, centre \(O\), \(OB\) is perpendicular to chord \(AC\).
Prove that a pair of triangles in this figure are congruent. (2 marks)
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\(\text{Proof (See Worked Solutions)}\)
\(\text{One of multiple proofs:}\)
\(OB\ \ \text{common} \)
\( \angle OBA = \angle OBC = 90^{\circ}\ \ \text{(given)} \)
\(OA = OC\ \ \text{(radii)} \)
\(\therefore\ \Delta AOB \equiv \Delta COB\ \ \text{(RHS)}\)
In the figure below, the line \(AD\) intersects \(BE\) at \(C\), \(BC = CD\) and \(AC = EC\).
Prove that this pair of triangles are congruent. (2 marks)
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\(\text{Proof (See Worked Solutions)}\)
\( \angle BCE = \angle DCE\ \ \text{(vertically opposite)} \)
\(BC = CD\ \ \text{(given)} \)
\(AC = EC\ \ \text{(given)} \)
\(\therefore\ \Delta ABC \equiv \Delta EDC\ \ \text{(SAS)}\)
The diagram shows two triangles that touch at the middle of a circle.
Prove that this pair of triangles are congruent. (2 marks)
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\(\text{Proof (See Worked Solutions)}\)
\(\text{Base of each triangle (chords) are equal} \)
\(\text{All other sides are equal radii of the circle} \)
\(\therefore\ \text{Two given triangles are congruent}\ \ \text{(SSS)}\)
The diagram shows two right-angled triangles where \(\angle BAC = \angle BDC = 90^{\circ}\), and \(AB = BD\).
Prove that this pair of triangles are congruent. (2 marks)
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\(\text{Proof (See Worked Solutions)}\)
\(BC\ \text{(hypotenuse) is common} \)
\(BA = BD\ \text{(given)} \)
\(\angle BAC = \angle BDC = 90^{\circ}\ \ \text{(given)} \)
\(\therefore \Delta ABC \equiv \Delta DBC\ \ \text{(RHS)}\)
The two triangles below are similar.
Find the length of \(ED\). (3 marks)
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\(ED = 15\)
\(\text{Using Pythagoras in}\ \Delta ABC: \)
\( AB=\sqrt{13^2-12^2}=\sqrt{25}=5 \)
\(\text{Scale factor}\ = \dfrac{FD}{AC} = \dfrac{39}{13} = 3 \)
\(\therefore ED = 3 \times AB = 3 \times 5 = 15 \)
A triangular prism is pictured below.
By what factor will its volume change if
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i. \(\text{Dimensions increase by a factor of 2}\)
\(\Rightarrow\ \text{Volume increases by a factor of}\ 2^3 = 8\)
ii. \(\text{Dimensions decrease by two-thirds}\)
\(\Rightarrow\ \text{i.e. adjust dimensions by a factor of}\ \ \dfrac{1}{3} \)
\(\Rightarrow\ \text{Volume decreases by a factor of}\ \Big{(} \dfrac{1}{3} \Big{)}^3 = \dfrac{1}{27} \)
Prove that the two triangles in the right cone pictured below are similar. (2 marks)
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\(\text{Proof (See Worked Solution)}\)
\(\angle ACD \ \text{is common to}\ \Delta ACE\ \text{and}\ \Delta BCD \)
\(\angle CAE=\angle CBD=90^\circ \ \text{(Right cone)} \)
\(\therefore \Delta ACE\ \text{|||}\ \Delta BCD\ \ \text{(equiangular)}\)
Show that \(\Delta ACD\) and \(\Delta DCB\) in the figure below are similar. (2 marks)
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\(\text{Proof (See Worked Solution)}\)
\(\angle BDC = 180-(88+44) = 48^{\circ} \ \text{(angle sum of Δ)} \)
\(\angle CAD = \angle CDB = 48^{\circ} \)
\(\angle ACD=44^{\circ} \ \text{is common to}\ \Delta ACD\ \text{and}\ \Delta BCD \)
\(\therefore \Delta ACD\ \text{|||}\ \Delta DCB\ \ \text{(equiangular)}\)
\(\text{Proof (See Worked Solution)}\)
\(\angle PQT = \angle SQR \ \text{(vertically opposite)} \)
\(\dfrac{PQ}{QS} = \dfrac{2.5}{5} = \dfrac{1}{2} \)
\(\dfrac{TQ}{QR} = \dfrac{2}{4} = \dfrac{1}{2} \)
\(\therefore \Delta PQT\ \text{|||}\ \Delta SQR\ \ \text{(sides adjacent to equal angles in proportion)}\)
Prove that this pair of triangles are similar. (2 marks)
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\(\text{Proof (See Worked Solution)}\)
\(\angle ACD = \angle BEA \ \text{(given)} \)
\(\angle BAE\ \text{is common to}\ \Delta ACD\ \text{and}\ \Delta BEA \)
\(\therefore \Delta ACD\ \text{|||}\ \Delta BEA\ \ \text{(equiangular)}\)
Prove that this pair of triangles are similar. (3 marks)
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\(\text{Proof (See Worked Solution)}\)
\(\text{Find unknown side}\ (x)\ \text{of smaller triangle}\)
\(\text{Using Pythagoras:}\)
\(x=\sqrt{5^2-4^2} = \sqrt{9} = 3\)
\(\dfrac{AC}{DE} = \dfrac{5}{15} = \dfrac{1}{3} \)
\(\dfrac{BC}{EF} = \dfrac{3}{9} = \dfrac{1}{3} \)
\(\angle ABC = \angle DEF = 90^{\circ} \ \ \text{(given)} \)
\(\therefore \Delta ABC\ \text{|||}\ \Delta DEF\ \ \text{(hypotenuse and second side of right-angled triangle in proportion)}\)
Prove that this pair of triangles are similar. (2 marks)
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\(\text{Proof (See Worked Solution)}\)
\(\dfrac{AB}{GH} = \dfrac{11}{22} = \dfrac{1}{2} \)
\(\dfrac{BC}{FG} = \dfrac{3.5}{7} = \dfrac{1}{2} \)
\(\angle ABC = \angle FGH = 95^{\circ} \ \ \text{(given)} \)
\(\therefore \Delta ABC\ \text{|||}\ \Delta HGF\ \ \text{(sides adjacent to equal angles in proportion)}\)
Prove that this pair of triangles are similar. (2 marks)
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\(\text{Proof (See Worked Solution)}\)
\(\angle ABE = \angle CBD\ \ \text{(vertically opposite)} \)
\(\angle AEB = \angle BCD\ \ \text{(alternate,}\ AE \parallel CD \text{)} \)
\(\therefore \Delta ABE\ \text{|||}\ \Delta DBC\ \ \text{(equiangular)}\)
Prove that this pair of triangles are similar. (2 marks)
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\(\text{Proof (See Worked Solution)}\)
\(\dfrac{RP}{AC} = \dfrac{18}{12} = \dfrac{3}{2} \)
\(\dfrac{QR}{BA} = \dfrac{12}{8} = \dfrac{3}{2} \)
\(\dfrac{PQ}{CB} = \dfrac{9}{6} = \dfrac{3}{2} \)
\(\therefore \Delta PQR\ \text{|||}\ \Delta CBA\ \ \text{(three pairs of sides in proportion)}\)