ADVANCED EXERCISE 08: QUADRILATERALS · Advanced Exercise 08: Quadrilaterals Page 13 25. The figure shows a triangle ABC and a parallelogram ADEC . F is a point on AC such that DF

Secondary 3 Mathematics Advanced Exercise 08: Quadrilaterals

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ADVANCED EXERCISE 08: QUADRILATERALS

1. The figure shows a rhombus ABCD with 20oACD∠ = . AC and BD intersect at E. F is a point outside the rhombus such that AEF is an equilateral triangle. Find DCF∠ .

2. In the figure, the diagonals of ABCD intersect at G. E and F are two points on BD such that BAE FCD∠ = ∠ . It is given that G is the mid-point of EF and EC // AF.

(a) Prove that AECF is a parallelogram. (b) Prove that ABCD is a parallelogram.


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3. In the figure, ABCD is a parallelogram. E, F, G and H are points on the four sides of ABCD such that AH = BE = CF = DG. Prove that EFGH is a parallelogram.

4. In the figure, ABCD and BEDF are two parallelograms. Two students, A and B, used two different approaches to prove that AE = CF.

(a) Student A got the answer by finding a triangle congruent to ADE∆ . Show the steps of student A.

(b) Student B got the answer by proving that AC and EF bisect each other. Show the steps of student B.


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5. In the figure, ABCD is a parallelogram. Q is a point outside the parallelogram. R and P are two points on AQ and BQ produced respectively such that AQ = QR and BQ = QP. Prove that PC and RD bisect each other.

6. The figure shows a parallelogram ABCD. P, Q, R and S are the mid-points of the 4 sides of ABCD. Each vertex is joined to a mid-point to form a quadrilateral A’B’C’D’ .

(a) Show that APCR is a parallelogram. (b) Hence show that A’B’C’D’ is a parallelogram. (c) Find AA’ : A’D : D’R. Hence find the ratio of the area of A’B’C’D’ to ABCD.


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7. The figure shows two square ABCD and AEFG. H is the intersection of DE and BG.

(a) Prove that ADE ABG∆ ≅ ∆ . (b) Hence prove that DE and BG are perpendicular.

8. (a) Figure (a) shows a parallelogram ABCD with AB = a, BC = b and ABC θ∠ = . Prove that the area of ABCD is

sinab θ .

(b) In figure (b), E is a point on the diagonal BD. Two lines XEY and KEL parallel to AB and BC respectively are

drawn. It is given that XE = x, YE = y, KE = k and LE = l . (i) Prove that xk = yl. (ii) Show that area of AKEX = area of LEYC.


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9. The figure shows a rectangle ABCD. AC and BD intersect at E. AEFG is a rhombus.

20oEAD∠ = and 50oEAG∠ = .

(a) Find GEB∠ . (b) Find CDF∠ .

10. In the figure, ABCD is a parallelogram. AE and BE are the angle bisectors of BAD∠ and ABC∠ respectively.

(a) Prove that 90oAEB∠ = . (b) EF is an altitude of AEB∆ . Find a pair of similar triangles with proof. (c) If AF = 9 and EF = 4, find the length of CD.


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11. In the figure, ABC∆ is a right-angled triangle. ACPD is a square.

(a) N is a point on AB such that the distance from P to AB is PN. Mark the point N on the figure and prove that PN AB BC= + .

(b) If the distance from P to AB is 8.5 cm and AQ = 6.5 cm, find the area of the polygon ABCPQ.

12. The figure shows a square ABCD, where E and F are the mid-points of AB and BC respectively.

(a) Prove that DE = DF. (b) G is a point outside the square such that DEGF is a rhombus. (i) Find GEB∠ . (ii) Explain whether G, B and D are collinear.


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13. In the figure, AH // IJC and HIB // DJ. BCF and DCG are straight lines. EG = FG and

CEF CEG∠ = ∠ . 90oAHB∠ = .

(a) Prove that CFE CGE∆ ≅ ∆ .

(b) Furthermore it is known that AB = BC = AD and 90oDAB∠ = . (i) Prove that ABCD is a square. (ii) Prove that IJ DJ JC= − . (iii) Hence find HJD∠ .

14. In the figure, AB = BD and AC = CF. E is the mid-point of DF.

(a) Prove that BCED is a parallelogram. (b) If the area of ABC∆ is 10 square units, find the area of the parallelogram BCED.


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15. In the figure, E, F, G and H are the mid-points of AB, AD, CD and BC respectively. Prove that EFGH is a parallelogram. (Hint: Join BD.)

16. In the figure, : 2 : 3AB BC= and BCEF is a parallelogram.

(a) Find :BD CE. (b) Find :GF FE .


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17. In the figure, AD is a median of ACB∆ . AP = PQ and AM = MD.

(a) Prove that BQ = AP. (b) E is a point on MD such that : 1 : 2ME ED= . CE cuts AB at N. Explain whether N is the mid-point of QP or

not.

18. The figure shows a parallelogram ABCD, with 2 diagonals AC and BD intersecting at K. P is the mid-point of AB. PC and BD intersect at Q.

(a) Show that 2BQ KQ= .

(b) Hence find :DQ QB.

(c) If the area of DQC∆ is 12 square units, find the area of APQD.


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19. In the figure, C is the mid-point of AE and AB // CD // EF. It is given that AB = x and EF = y.

(a) Find the length of CD in terms of x and y. (b) Find the length of GH in terms of x and y.

20. In the figure, PQRS is a rectangle with PQ = 10 cm. When the rectangle is rotated through an angle θ clockwise about S to form P’Q’R’S’, the distance between P’ and Q will be the shortest and is equal to 2 cm.

(a) (i) By drawing a figure to show the situation, find the length of PS. (ii) Find the value of θ . (iii) When the distance between P’ and Q is the shortest, how many intersection between rectangles PQRS and

P’Q’R’S are there? (b) Let the distance between P’ and Q be d cm. Find the possible range of values of d when the rectangle rotates.


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21. In the figure, N is a point such that AN is an angle bisector of ABC∆ and also an altitude of ABK∆ . D is the mid-point of BC. AC and BN are produced to meet each other at K.

(a) (i) Prove that AB = AK. (ii) Prove that DN // AK.

(b) Prove that ( )1

2DN AB AC= − .

22. In the figure, AB // CD // EF and AC = CE. AB = 8 cm and EF = 12 cm.

(a) Find the length of CD. (b) If the area of ABDC is 18 cm2, find the area of CDFE.


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23. In the figure, BCD // EFG. If is given that AB = 1, BE = 2, DG = 3 and GH = 4. ACFH is a straight line.

(a) Find :AC CF . (b) Hence find the ratio : :AC CF FH .

24. In the figure, BC // DEF. AB = BD and DE = EF.

(a) Find :AC CE and :BC DE. (b) Prove that ~BCQ FED∆ ∆ . Hence find :AQ QE.


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25. The figure shows a triangle ABC and a parallelogram ADEC. F is a point on AC such that DF // BGC.

(a) Prove that ADF ECG∆ ≅ ∆ . (b) Name all triangles which are similar to BDG∆ . (c) Now : 1 :BD DA m= and the area of BDG∆ is 1 square unit. Find the area of the following in terms of m. (i) DACG (ii) BACE

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ADVANCED EXERCISE 08: QUADRILATERALS · Advanced Exercise 08: Quadrilaterals Page 13 25. The figure shows a triangle ABC and a parallelogram ADEC . F is a point on AC such that DF