HONG KONG DIPLOMA OF SECONDARY EDUCATION
EXAMINATION
MATHEMATICS
Compulsory Part
SCHOOL-BASED ASSESSMENT
Sample Assessment Task
Figures Inscribed in a Right-angled Triangle
Marking Guidelines
教育局 課程發展處 數學教育組
Mathematics Education Section, Curriculum Development Institute
The Education Bureau of the HKSAR
Assessment Scale
The assessment scale for tasks on Mathematical Investigation is shown in the following table.
Level of
Performance
Very good
Good
Fair
Weak
Marks
Mathematical Knowledge and Investigation Skills
The student demonstrates a complete understanding of
the underlying mathematical knowledge and
investigation skills which are relevant to the task, and is
13–16 consistently competent and accurate in applying them in
handling the task. Typically, the student recognises
patterns, and states a conjecture which is justified by a
correct proof.
9–12
The student demonstrates a substantial understanding of
the underlying mathematical knowledge and
investigation skills which are relevant to the task, and is
generally competent and accurate in applying them in
handling the task. Typically, the student recognises
patterns, draws inductive generalisations to arrive at a
conjecture and attempts to provide a proof for the
conjecture.
5–8
The student demonstrates a basic understanding of the
underlying mathematical knowledge and investigation
skills which are relevant to the task, and is occasionally
competent and accurate in applying them in handling
the task. Typically, the student recognises patterns and
attempts to draw inductive generalisations to arrive at a
conjecture.
1–4
The student demonstrates a limited understanding of the
underlying mathematical knowledge and investigation
skills which are relevant to the task, and is rarely
competent and accurate in applying them in handling
the task. Typically, the student manages to recognise
simple patterns and organises results in a way that helps
pattern identifications.
Marks Mathematical Communication Skills
4
The student communicates ideas in
a clear, well organised and logically
true manner through coherent
written/verbal accounts, using
appropriate and correct forms of
mathematical
presentation
to
present conjectures and proofs.
3
The student is able to communicate
ideas
properly
through
written/verbal accounts, using
appropriate and correct forms of
mathematical presentation such as
algebraic
manipulations
and
geometric deductions.
2
The student is able to communicate
ideas with some appropriate forms
of mathematical presentation such
as
algebraic
formulae
and
geometric facts.
1
The
student
attempts
to
communicate
ideas
using
mathematical
symbols
and
notations,
diagrams,
tables,
calculations etc.

The full mark of a SBA task on Mathematical Investigation submitted should be scaled to 20 marks, of which 16 marks are
awarded for the mathematical knowledge and investigation skills while 4 marks are awarded for the communication skills.

Teachers should base on the above assessment scale to design SBA tasks for assessing students with different abilities and to
develop the marking guidelines.
Figures Inscribed in a Right Angled Triangle Marking Guidelines 2
Marking Guidelines
Solution
Part A
1.
(a)
c2  a 2  b2 .
(b)
 APS = ABR
ASP = ARB
PAS = BAR
 APS ~ ABR
(corr s, PS // BR)
(corr s, PS // BR)
(common)
Evidence:
1.
The relation between a , b and c
2.
The proof
3.
The deduction
Weak:
Attempt to give the relation,
proof and deduction, but
they are all wrong
Fair:
Able to give the relation,
and complete the proof
correctly without reasons
Good:
Able to give the relation,
and complete the proof
correctly with reasons
(AAA)
 APS ~ ABR
x
bx


a
b
bx  ab  ax
x
Performance
ab
ab
Very Good: Able to give the relation,
complete the proof correctly
with reasons and make
correct deduction
Part B
1.
Let O be the centre of the inscribed circle.


Evidence:
The proof
Area of OAB + Area of OAR + Area of OBR = Area of ABR
ra rb rc ab
  
2 2 2
2
r  a  b  c   ab
r
ab
abc
Weak:
Attempt to do the proof,
but fail to find the relation
between a , b , c and r
Fair:
Attempt to use areas to do
the proof but fail to get the
correct relation between
a , b , c and r
Good:
Attempt to use areas to do
the proof and get the
correct relation between
a , b , c and r. But, minor
mistakes are found in the
proof
Very Good: The proof is correct
Figures Inscribed in a Right Angled Triangle Marking Guidelines 3
Marking Guidelines
Solution
Part C
1.
(a)
Using the result in Part A,
6  9 18

69 5
1 18 9
 The radius of the inscribed circle    units.
2 5 5
Performance
Evidence:
1.
The solution
2.
The proof
length of the side of the square PQRS 
(b)
Let r be the radius of the circumcircle.
PR 2  PS 2  SR 2
2
 2r 2   18    18 
 5  5
18
2r 
2
5
9
r
2 units
5
(c)
Area of the shaded region
2
2
1  9
  18   1
2     
=  
4  5
  5   2
2
Weak:
Fail to
answers
give
correct
Fair:
Able to get one correct
answer with valid working
Good:
Able to get all the correct
answers
with
valid
working
Very Good: Able to get all the correct
answers
with
valid
working, and the correct
proof
2
 18 2
9 
      
5 
 5 
182
100
 3.24 sq units
=
Part D
Two examples of constructing the two identical circles inside triangle DEF
are given below:
Evidence:
1.
The radius of circle
2.
The procedures
Example 1
Two identical circles, with each one touching 2 sides of the triangle DEF,
are given in Figure 5. The radius of the circle is 2 units.
Procedures
(a) Draw a perpendicular line from D to EF meeting EF at R.
(b)
Locate the centre of each circle.
ER = RF (properties of isosceles triangle)
DER and DRF are two congruent right-angled triangles.
10
2
2
ER  RF   5 , DR  13   5   12
2
12  5
Using the result in Part B, radius of the inscribed circle is
2.
12  13  5
Construct a square OSRT with the length of each side 2 units.
O is the centre of the inscribed circle.
The other centre of the inscribed circle can be located by similar method.
(c)
Construct the inscribed circles of DER and DRF .
Figures Inscribed in a Right Angled Triangle Marking Guidelines 4
Weak:
Fail to find the radius
Fair:
Able to find the radius
Good:
Able to find the radius but
the procedures are not
comprehensive
Very Good: Able to find the radius and
give
comprehensive
procedures
Marking Guidelines
Solution
Performance
D
13
13
O
2
E
2
Figure 5
S
2
T 2
F
R
5
5
Example 2
Two identical circles, with one circle touching 2 sides of the triangle DEF
and the other touching only one side, are given in Figure 6.
D
Figure 6
E
Figures Inscribed in a Right Angled Triangle Marking Guidelines 5
F