11 pp. 82 – 93
Level 1
1. corr. Ðs, PQ // BC
alt. Ðs, PQ // BC
given
\
\ sides opp. eq. Ðs
\ DBCP is an isosceles triangle.
2.
\
\
\ OB is the altitude of DOBD.
3. (a) given
\ corr. Ðs, @ Ds
\ AD is the angle bisector of ÐBAC.
(b) given
\ corr. sides, @ Ds
corr. Ðs, @ Ds
adj. Ðs on st. line
\
\
\ AD is the perpendicular bisector of BC.
4. (a) In and ,
vert. opp. Ðs
Ð sum of D
\ AAA
(b)
\ (corr. sides, ~ Ds)
cm
5. In DQPM and DRPM,
given
common side
\ ASA
\ corr. sides, @ Ds
\ PM is the median of DPQR.
6. According to triangle inequality,
\
\
From (1) and (2),
\ x = 7, 8, 9
(or other reasonable answers)
7. In DABX,
(triangle inequality)
In DABC,
(triangle inequality)
(1) + (2):
8. G is the centroid of DABC.
\
\
9. Step 1:
In, use A as a centre to draw an arc on BC that meets the opposite side at P and Q respectively.
Step 2:
Use P and Q as centres with equal radii to draw 2 arcs on a side of BC at R. Join AR which is an altitude of the triangle.
Step 3:
Repeat the previous steps with B and C as centres to draw the remaining 2 altitudes of.
The point of intersection of the 3 altitudes is the orthocentre of the triangle.
10. (a) P is the in-centre of DABC.
(b) In DABC,
(Ð sum of D)
11. (diags. of // gram)
(diags. of // gram)
12. (a)
\ (alt. Ðs, AD // BC)
\ (alt. Ðs, AB // DC)
(vert. opp. Ðs)
\ , and
\ , and
(sides opp. eq. Ðs)
\ DABP, DDQP and DCQB are isosceles triangles.
(b) (opp. sides of // gram)
is an isosceles triangle.
\
13.
\ (int. Ðs, AD // BP)
DC is the angle bisector of ÐADP.
\
\ (int. Ðs, AB // DC)
14. (a) In and ,
\ alt. Ðs, BP // SD
Ð sum of D
\ AAA
(b)
\ (corr. Ðs, ~ Ds)
\
\ (int. Ðs, AD // BC)
(int. Ðs, AD // BC)
\ and
\ ABCD is a parallelogram. (opp. Ðs equal)
15. (a)
\ (alt. Ðs equal)
\ ABFE is a parallelogram.
(opp. sides equal and //)
(b) (opp. sides of // gram)
and
\ BCDE is a parallelogram.
(diags. bisect each other)
16. (property of rhombus)
AP is the angle bisector of ÐBAC.
\
(property of rhombus)
In DAPC,
(ext. Ð of D)
17. (a) In DBCP,
(Ð sum of D)
\
(int. Ðs, AD // BC)
(b) (property of rectangle)
18. (a) (opp. sides of rectangle)
(opp. sides of // gram)
\
(b) In DPCQ,
(property of rectangle)
(proved)
\ (intercept theorem)
and
\
\
19. (a) (property of square)
(property of equil. D)
\
\ DBQR is an isosceles triangles.
(b) (property of square)
(property of equil. D)
In DBQR,
\ (base Ðs, isos. D)
(Ð sum of D)
(property of square)
20. ABCD is a isosceles trapezium.
\ and
In and ,
proved
given
proved
\ AAS
\ corr. sides, @ Ds
21. (a) (property of rectangle)
(opp. sides of // gram)
\
(b) In and ,
proved
property of square
opp. sides of // gram
\ SSS
22. As shown in the figure, extend EF to intersect with BC at G.
ABCD is a trapezium.
\
\
\ ABGF is a parallelogram.
\ (opp. sides of // gram)
(opp. sides of // gram)
\
\ (intercept theorem)
23. AP = PB and AR = RC
\ (mid-pt. theorem)
BP = PA and BQ = QC
\ (mid-pt. theorem)
CQ = QB and CR = RA
\ (mid-pt. theorem)
\ The perimeter of DABC is 80 cm.
24. As shown in the figure, join AD such that AD and BE intersect at G.
and
\ (intercept theorem)
and
\ (mid-pt. theorem)
and
\ (intercept theorem)
and
\ (mid-pt. theorem)
25. and
\ (intercept theorem)
and
\ (intercept theorem)
\
Level 2
26. BX is the perpendicular bisector of OA.
\
In and ,
proved
given
common side
\ SAS
\ corr. Ðs, @ Ds
\ base Ðs, isos. D
\ is an isosceles triangle.
Similarly, is an isosceles triangle.
and
(Ð sum of D)
\
27. (a) BE and CD are the medians of DACE.
\ and
\ (mid-pt. theorem)
\ (corr. Ðs, BD // CE)
\ (corr. Ðs, @ Ds)
(base Ðs, isos. D)
\
Hence CE is the angle bisector of ÐAEF.
(b)
\
\
\ (base Ðs, isos. D)
\
\ (alt. Ðs equal)
28. (a) In DABS and DARS,
given
common side
given
\ ASA
\ corr. sides, @ Ds
(b) In DBCR,
and
\ (mid-pt. theorem)
(c) (Pyth. theorem)
\ (corr. sides, @ Ds)
In DBCR,
and
\ (mid-pt. theorem)
\
29. In DACD and DABE,
According to triangle inequality,
(1) + (2):
\ ,
\
\
30. (a) As shown in the figure, construct a line SR such that SR // AB, where S is a point on line AC.
In DABC ,
and
\ (intercept theorem)
(corr. Ðs, RS // BA)
(adj. Ðs on st. line)
\
In and ,
common side
proved
proved
\ SAS
\ corr. sides, @ Ds
(b) (proved)
\ (base Ðs, isos. D)
(Ð sum of D)
(Ð sum of D)
\
\
AQ is the angle bisector of ÐBAC.
\
\
31. (a)
(b)
(c) Yes, centroid.
32. (a) (opp. sides of // gram)
(opp. sides of // gram)
From (1),
Substitute (3) into (2):
Substitute x = 8 into (3):
(b)
\ Perimeter of ABCD
33. (a) In and ,
\
alt. Ðs, AD // BC
given
\ SAS
(b)
\ (corr. sides, @ Ds)
(corr. sides, @ Ds)
(adj. Ðs on st. line)
(adj. Ðs on st. line)
\
\ (alt. Ðs equal)
(c) and
\ PQRS is a parallelogram.
(opp. sides equal and //)
34. (a) In and ,
and
\
(property of rhombus)
(property of rhombus)
\ (AAS)
(b)
\ (corr. Ðs, @ Ds)
(property of rhombus)
\
\ AC is the angle bisector of ÐPCQ.
35. (a) In and ,
PQR is the perpendicular bisector of BD.
\
given
\ SAS
\ corr. sides, @ Ds
\ DBPD is an isosceles triangles.
(b) In DBRQ and DDPQ,
alt. Ðs, AD // BC
given
vert. opp. Ðs
\ ASA
(c)
\
\ (corr. sides, @ Ds)
(d) (alt. Ðs, AD // BC)
(Ð sum of D)
36. (a) In and ,
alt. Ðs, PS // QR
alt. Ðs, PS // QR
vert. opp. Ðs
\ AAA
(b) is an isosceles trapezium.
\ and
In DPQR and DSRQ,
given
given
\ SAS
\ corr. sides, @ Ds
37. (a) property of square
given
\
property of square
\
\ BCSD is a
parallelogram. opp. sides equal and //
(b)
\ (base Ðs, isos. D)
\ (alt. Ðs, AD // BC)
\ (alt. Ðs, BD // CS)
\
\ (sides opp. eq. Ðs)
(c) (property of square)
(proved)
\
(int. Ðs, BD // CS)
(proved)
(property of square)
\
(Ð sum of D)
(adj. Ðs on st. line)
(d) (corr. Ðs, AB // DC)
(base Ðs, isos. D)
\
\ (sides opp. eq. Ðs)
38. (a) In and ,
(corr. Ðs, BA // TS)
(corr. Ðs, BA // TS)
(Ð sum of D)
\ (AAA)
(b) and
\ (intercept theorem)
and
\ (mid-pt. theorem)
In DSCT,
(Pyth. theorem)
(corr. sides, ~ Ds)
39. (a) AB, PQ and DC are perpendicular to BC.
\
\ AB // PQ (corr. Ðs equal)
PQ // DC (int. Ðs supp)
\ AB // PQ // DC
\ (intercept theorem)
In and ,
common side
proved
\ SAS
\ corr. sides, @ Ds
(b) Construct a line ARS such that ARS // BC, where ARS intersects PQ and DC at R and S respectively.
ABQR and ABCS are rectangles.
\
and
\ (intercept theorem)
and
\ (mid-pt. theorem)
40. (a)
\
\ (intercept theorem)
and
\ (mid-pt. theorem)
(opp. sides of // gram) Ds)
\
\
(b) and
\ (mid-pt. theorem)
(c) and
\ (intercept theorem)
, and
\
\
In DBCQ and DPCQ ,
(given)
(proved)
(common side)
\ (SSS)
(d)
\ (corr. Ðs, @ Ds)
(adj. Ðs on st. line)
\
\
\
Multiple-choice Questions
1. A
Let , .
(Ð sum of D)
In ,
(Ð sum of D)
\
\ BP is the altitude of DABC.
2. D
(SAS)
\ AB = AC (corr. sides, @ Ds)
3. D
(Pyth. theorem)
4. D
In DPQR,
(Ð sum of D)
\ PQ // SR (int. Ðs, supp.)
(ext. Ð of D)
\ PR ^ QS
5. D
Let , .
In DABC,
(ext. Ð of D)
In DBCD,
(ext. Ð of D)
6. A
7. B
In DABC and DDBA,
(given)
(common Ð )
(Ð sum of D)
\ (AAA)
\
8. A
From (1),
(2) + (3):
\
9. D
The positions of the circumcentre and the centroid of an equilateral triangle are the same.
OR : OC = 1 : 2
\
10. C
11. B
12. D
According to triangle inequality,
\
\ and
13. D
(property of equil. D)
and
\ and
(property of isos. D)
14. D
15. B
(1) – (2):
Substitute x = 3 into (2):
16. C
In and ,
(vert. opp. Ðs)
(alt. Ðs, AD // BC)
(alt. Ðs, AD // BC)
\
\
\
17. A
In DBCE and DDAG,
(alt. Ðs, BC // AD)
(opp. sides of // gram)
(given)
\ (ASA)
\ (corr. sides, @ Ds)
(corr. Ðs, @ Ds)
(adj. Ðs on st. line)
(adj. Ðs on st. line)
\
\ (alt. Ðs equal)
\ AECG is a parallelogram. (opp. sides equal and //)
18. A
(property of equil.D)
(alt. Ðs, AD // BC)
19. B
(property of rhombus)
20. C
(property of equil.D)
(property of square)
\
(property of equil.D)
(property of square)
\
\ (base Ðs, isos. D)
(Ð sum of D)
21. A
ABFE is a rectangle.
\
(property of rhombus)
\ (property of rhombus)
22. D
As shown in the figure, construct a line SD such that BA // SD, where SD intersects PQ and BC at R and S respectively.
APRD and PBSR are parallelograms.
\ (opp. sides of // gram)
and
\ (mid-pt. theorem)
Perimeter of ABCD
23. C
As shown in the figure, construct QS and RT such that QS // AB and RT // AC, where S and T are the points on line BP.
In and ,
(corr. Ðs, QS // AB)
(corr. Ðs, RT // AC)
(Ð sum of D)
\ (AAA)
and
\ (intercept theorem)
and
\ (mid-pt. theorem)
\ (corr. sides, ~ Ds)
and
\ (intercept theorem)
and
\ (mid-pt. theorem)
\
24. B
and
\ (intercept theorem)
and
\ (intercept theorem)
Substitute y = 6 into (1):
25. A
and
\ (intercept theorem)
and
\ (mid-pt. theorem)
and
\ (intercept theorem)
\ and
\ (mid-pt. theorem)
\
55
© Hong Kong Educational Publishing Co.