9th Mathematics Science All Chapter Annual Questions-I-2019-2020

Download Tamil Nadu 9th Standard Maths question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.

Questions:

Generic 2

14 x 2 = 28

Find the value of \(\cfrac { cos{ 63 }^{ 0 }20' }{ sin{ 26 }^{ 0 }40' } \)

Given that \(sin\alpha =\cfrac { 1 }{ \sqrt { 2 } } \) and \(tan\beta =\sqrt { 3 } \) Find the value of \(\alpha +\beta \)

The outer dimensions of a closed wooden box are 10 cm by 8 cm by 7 cm, Thickness of the wood is 1 cm³. Find the total cost of wood required to make box if 1 cm³ of wood costs Rs. 2.00?

Find the centroid of the triangle whose vertices are (2, -5), (5, 11) and (9, 9)

If A (10, 11) and B (2 ,3) are the coordinates of end points of diameter of circle. Then find the centre of the circle.

In a football match, a goalkeeper of a team can stop the goal, 32 times out of 40 attempts tried by a team. Find the probability that the opponent team can convert the attempt into a goal.

Find any 3 irrational numbers between 0.12 and 0.13.

Convert the following rational numbers into decimal
(i) \(2\over 3\)
(ii) \(47\over 99\)
(iii) \(-{16\over 45}\)

Find the value of \({ 729 }^{ \frac { -5 }{ 6 } }\)

10.

A set of numbers consists of five 4’s, four 5’s, nine 6’s,and six 9’s. What is the mode.

11.

Determine whether (x -1) is a factor of the following polynomials: x³+5x²-10x+4

12.

Plot the points A( -1, 0), B( 3, 0), C(3, 4), and 0(-1, 4) on a graph sheet. Join them to form a rectangle. Draw the mirror image Of the diagram in clockwise direction:
(i) about x-axis.
(ii) about y-axis.
What is your observation on the coordinates of the mirror image?

13.

Write down the power set of the following sets.
A = {a, b}

14.

Identify which ones are trapeziums and which are not.

Generic 5

3 x 5 = 15

15.

Solve by the method of elimination
(i) 2x–y = 3; 3x + y = 7
(ii) x–y = 5; 3x + 2y = 25
(iii) \(\frac { x }{ 10 } +\frac { y }{ 5 } =14;\frac { x }{ 8 } +\frac { y }{ 6 } =15\)
(iv) 3(2x + y) =7xy; 3(x + 3y) = 11xy
(v) \(\frac { 4 }{ x } +5y=7;\frac { 3 }{ x } +4y=5\)
(vi) 13x +11y = 70; 11x +13y = 74

16.

Find the quotient and remainder when 5x³ + 7x² + 3x + 2 is divided by 3x + 2

17.

In the given Fig. if AB = 2, BC = 6, AE = 6, BF = 8, CE = 7, and CF = 7, compute the ratio of the area of quadrilateral ABDE to the area of ΔCDF(Use congruent property of triangles).

Generic 3

13 x 3 = 39

18.

The probability that it will rain tomorrow is \(\\ \frac { 91 }{ 100 } \). What is the probability that it will not rain tomorrow?

19.

Find the TSA and LSA of a cuboid whose length, breadth and height are 7.5 m, 3 m and 5 m respectively.

20.

Express
(i) sin 74° in terms of cosine
(ii) tan 12° in terms of cotangent
(iii) cosec 39° in terms of secant

21.

Show that (x-3) is a factor of x³+ 9x²- x - 105

22.

Add \(\sqrt [ 5 ]{ 11 } \) and \(\sqrt [ 7 ]{ 11 } \). Check whether the sum is rational or irrational

23.

A researcher studying the behaviour of mice has recorded the time (in seconds) taken by each mouse to locate its food by considering 13 different mice as 31, 33, 63, 33, 28, 29, 33, 27, 27, 34, 35, 28, 32.Find the median time that mice spent in searching its food.

24.

If \(\sqrt{2}\) =1.414, \(\sqrt{3}\) = 1.732, \(\sqrt{5}\) = 2.236, \(\sqrt{10}\) = 3.162 then find the values of the following correct to 3 places of decimals.
(i) \(\sqrt { 40 } -\sqrt { 20 } \)
(ii) \(\sqrt { 300 } -\sqrt { 90 } -\sqrt { 8 } \)

25.

If U = {x: −4 ≤ x ≤ 4, x \(\in \)Z}, A = {x :−4 < x ≤ 2, x\(\in \)Z } and B = {x :−2 ≤ x ≤ 3,x \(\in \) Z} , then verify De Morgan’s laws for complementation.

26.

Study the following pattern and write the algebraic expression
(i)

Shape Number	1	2	3	4	5
Number of matchsticks	4	7	10	13	16

(ii)

Shape Number	1	2	3	4	5
Number of Square Boxes	1	4	7	10	13

27.

In a class there are 40 students. 26 have opted for Mathematics and 24 have opted for Science. How many student have opted for Mathematics and Science.

28.

Join the points taken in order and name the figure obtained. Name atleast any one point which is lying :
(i)On x-axis
(ii)On y-axis
(iii) In each of the Quadrants.

29.

Express the following decimal expression into rational numbers -5.132

30.

The point (x, y) is equidistant from the points (3, 4) and (–5, 6). Find a relation between x and y.

Multiple Choice Question

14 x 1 = 14

31.

The six faces of the dice are called equally likely if the dice is _______.

(a)

Small

(b)

Fair

(c)

Six-faced

(d)

Round

32.

If the sides of a triangle are 3 cm, 4 cm and 5 cm, then the area is _______.

(a)

3 cm²

(b)

6 cm²

(c)

9 cm²

(d)

12 cm²

33.

if sin 30⁰ = x and cos 60⁰ = y, then x² + y² is________.

(a)

\(\frac { 1 }{ 2 } \)

(b)

(c)

sin90°

(d)

cos90°

34.

The centroid of the triangle with vertices (−1, −6), (−2, 12) and (9, 3) is

(a)

(3, 2)

(b)

(2, 3)

(c)

(4, 3)

(d)

(3, 4)

35.

In a town 30% like only coffee, 20% like only tea,10% like only like,15% like only any two of them, 5% only like all the three. What is the percentage of people who like none them.

(a)

15%

(b)

20%

(c)

10%

(d)

36.

Degree of the linear polynomial is ________

(a)

(b)

(c)

(d)

37.

In simple form,\(\sqrt [ 3 ]{ 54 } \) is ?

(a)

\(3\sqrt [ 3 ]{ 2 } \)

(b)

\(3\sqrt { 27 } \)

(c)

\(3\sqrt { 2 } \)

(d)

\(\sqrt { 3 } \)

38.

The mean of 5, 9, x, 17,and 21 is 13 then find the value of x ___________

(a)

(b)

(c)

(d)

39.

The mean of the first 10 prime numbers is ___________

(a)

12.6

(b)

12.7

(c)

12.8

(d)

12.9

40.

The length and breadth of a rectangular plot are 5 \(\times\) 10⁵ and 4 \(\times\) 10⁴ metres respectively. Its area is ______.

(a)

9 \(\times\) 10¹ m²

(b)

9 \(\times\) 10⁹ m²

(c)

2 \(\times\) 10¹⁰ m²

(d)

20 \(\times\) 10²⁰ m²

41.

If n(A \(\cup \) B \(\cup \) C) = 40, n(A) = 30, n(B) = 25, n(C) = 20, n(A\(\cap \)B) = 12, n(B\(\cap \)C) = 18 and n(A\(\cap \)C) = 15 , then n(A\(\cap \)B\(\cap \)C) is ___________

(a)

(b)

(c)

(d)

42.

The point of concurrency of the medians of a triangle is known as __________

(a)

circumcentre

(b)

incentre

(c)

orthocentre

(d)

centroid

43.

The centre of a circle is (0, 0). One end point of a diameter is (5, -1), then ______________

(a)

\(\sqrt{24}\)

(b)

\(\sqrt{37}\)

(c)

\(\sqrt{26}\)

(d)

\(\sqrt{17}\)

44.

x³ – x² is a …………..

(a)

monomial

(b)

binomial

(c)

trinomial

(d)

constant polynomial

Generic 8

2 x 8 = 16

45.

Draw and locate the centroid of the triangle ABC where right angle at A, AB = 8 cm and AC = 6 cm.

46.

Draw the \(\triangle \)ABC , where AB = 6 cm, B = 110° and AC = 9 cm and construct the centroid.

Visit QB365-SLMS - https://www.qb365.in/slms/

Answers Key:

Generic 2

14 x 2 = 28

\(\cfrac { cos{ 63 }^{ 0 }20' }{ sin{ 26 }^{ 0 }40' } =\cfrac { cos63^{ 0 }20' }{ cos{ 63 }^{ 0 }20' } =1\)

\(\alpha ={ sin }^{ -1 }\left( \cfrac { 1 }{ \sqrt { 2 } } \right) ={ 45 }^{ 0 }\)
\(\beta \) = 60⁰
\(\alpha +\beta ={ 105 }^{ 0 }\)

Volume of wood = 12 \(\times\) 10\(\times\) 9 - 10 \(\times\) 8 \(\times\) 7
= 1080 - 560 = 520 cm³ \(\times\)2.00 = Rs. 1040

\(G\left( \frac { { x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 } }{ 3 } ,\frac { { y }_{ 1 }+{ y }_{ 2 }+{ y }_{ 3 } }{ 3 } \right) =\left( \frac { 2+5+9 }{ 3 } ,\frac { -5+11+9 }{ 9 } \right) =\left( \frac { 16 }{ 3 } ,5 \right) \)

Centre of the circle \(=\left( \frac { 10+2 }{ 2 } ,\frac { 11+3 }{ 2 } \right) =(6,7)\)

Total no. of attempts n(S) = 40
Total no. of attempts by A team n(A) = 32
Total no. of attempts by the opponent team B = n(B) = 40 - 32 = 8
P(B) = \(\frac { n(B) }{ n(S) } =\frac { 8 }{ 40 } =\frac { 1 }{ 5 } \).

Three irrational numbers between 0.12 and 0.13 are 0.12010010001…, 0.12040040004…, 0.12070070007…

(i) \(2\over 3\) = 0.666... = \(0.\overline 6\)

(ii) \(47\over 99\) = 0.4747... = \(0.\overline {47}\)

(iii) \(-{16\over 45}\) = 0.355.... = \(0.3\overline{5}\)

\({ 729 }^{ \frac { -5 }{ 6 } }\) =\(\cfrac { 1 }{ { 729 }^{ \cfrac { 5 }{ 6 } } } =\cfrac { 1 }{ \left( \sqrt [ 6 ]{ 729 } \right) ^{ 5 } } =\cfrac { 1 }{ \left( \sqrt [ 6 ]{ { 3 }^{ 6 } } \right) } =\cfrac { 1 }{ { 3 }^{ 5 } } =\cfrac { 1 }{ 243 } \)

10.

Size of item	4	5	6	9
Frequency	5	4	9	6

6 has the maximum frequency 9. Therefore 6 is the mode.

11.

Let P(x) = x³+5x²-10x+4
By factor theorem (x-1) is a factor of P(x), if P(1) = 0
P(1) = 1³+5(1²)-10(1)+4
= 1+5-10+4
P(1) = 0
ஃ (x-1) is a factor of x³+5x²-10x+4

12.

Take a graph sheet and plot all the points and join the points to form a rectangle.
Take a mirror and see the image of the rectangle formed.

13.

{{ }, {a}, {b} {a, b}}

14.

It is not a trapezium

Generic 5

3 x 5 = 15

15.

(i) 2x - y = 3 ....(1)
3x + y = 7 ...(2)
(1)+(2) \(\Rightarrow\) 2x - y = 3
\(\cfrac { 3x+y=7 }{ 5x=10 } \)
\(x=\cfrac { 10 }{ 5 } =2\)
Substitute x = 2 in (1)
2(2) -y = 3
4-y = 3
-y = 3 - 4
-y = -1
\(\therefore\) x = 2; y = 1
(ii) x - y = 5 ....(1)
3x + 2y = 25 ...(2)
(1) \(\times\) 3 \(\Rightarrow\) 3x - 3y = 15 ...(3)
(2) \(\Rightarrow \cfrac { 3x+2y=25 }{ -5y=-10 } \)
(3) - (2)
y = 2
Substitute y = 2 in (1)
x - 2 = 5
x = 5 + 2
x = 7
\(\therefore\) x = 7, y = 2
(iii) \(\cfrac { x }{ 10 } +\cfrac { y }{ 5 } =14\Rightarrow \cfrac { x+2y }{ 10 } =14\)
x + 2y = 140 .......(1)
\(\cfrac { x }{ 10 } +\cfrac { y }{ 6 } =15\Rightarrow \cfrac { 3x+4y }{ 24 } =15\)
3x + 4y = 360 ......... (2)

Substitute y = 30 in (1)
x + 2 (30) = 140
x + 60 = 140
x = 140 - 60
x = 80
\(\therefore\) x = 80; y = 30
(iv) 3(2x + y) = 7xy \(\Rightarrow\) 6x + 3y = 7xy
3(x + 3y) = 11xy \(\Rightarrow\) 3x + 9y = 11xy

\(\cfrac { 6 }{ y } +\cfrac { 3 }{ x } =7\)

\(\cfrac { 3 }{ y } +\cfrac { 9 }{ x } =11\) ....(4)
In (3), (4) Put \(\cfrac { 1 }{ x } =a,\cfrac { 1 }{ y } =b\)
(3) \(\Rightarrow\) 36b + 3a = 7
(4) \(\Rightarrow\) 3b +9a = 11
(6) X (2) \(\Rightarrow\) 6b + 18a = 22
(5) \(\Rightarrow\) \(\cfrac { 6b+3a=7 }{ 15a=15 } \)
(3)-(2)
\(a=\cfrac { 15 }{ 15 } \)
Substitute a = 1 in (5)
6b + 3 (1) = 7
6b + 3 = 7
6b = 7 - 3
\(b=\cfrac { 4 }{ 6 } =\cfrac { 2 }{ 3 } \)
\(\therefore \ a=\cfrac { 1 }{ x } =1\Rightarrow x=1\)
\(b=\cfrac { 1 }{ y } =\cfrac { 2 }{ 3 } \Rightarrow y=\cfrac { 3 }{ 2 } \)
\(\therefore\) Solution x = 1; \(y=\cfrac { 3 }{ 2 } \)
(iv) Put  \(\cfrac { 1 }{ x } =a\)
\(\cfrac { 4 }{ x } +5y=7\Rightarrow 4a+5y=7\) ....(1)
\(\cfrac { 3 }{ x } +4y=5\Rightarrow 3a+4y=5\) ...(2)
(1) \(\times\) 3 \(\Rightarrow\) 12a +15y = 21
(2) 4 \(\Rightarrow\) \(\cfrac { 12a+16y=20 }{ -1y=1 } \)
y = 1
Substitute y = -1 in (1)
4a + 5 (-1)=7
4a - 5 = 7
4a = 7 + 5
\(a=\cfrac { 12 }{ 4 } =3\)
\(a=\cfrac { 1 }{ x } =3\Rightarrow x=\cfrac { 1 }{ 3 } \)
\(\therefore\) y = -1
\(\therefore\)  \(x=\cfrac { 1 }{ 3 } \) ; y = -1
(vi) 13x + 11y = 70 --------- (1)
11x + 13y = 74 --------- (2)
(1) \(\times\) 11 \(\Rightarrow\) 143x+121y = 770 .... (3)
(2) \(\times\) 13 \(\Rightarrow\) \(\cfrac { 143x+169y=962 }{ -48y=-192 } \) ....(4)
(3) - (4)
\(y=\cfrac { 192 }{ 48 } =4\)
Substitute y = 4 in (1) 48
13x + 11 (4) = 70
13x + 44 = 70
13x = 70 - 44 = 26
\(x=\cfrac { 26 }{ 13 } =2\)
\(\therefore\)  x = 2; y = 4

16.

P (x) = 5x³ + 7x² + 3x + 2
d(x) = 3x + 2
Standard form of p (x) = 5x³+ 7x² + 3x + 2
and d (x) = 3x + 2

5x³ + 7x² + 3x + 2 = \(\left( x+\cfrac { 2 }{ 3 } \right) \left( 5{ x }^{ 2 }-{ \cfrac { 11 }{ 3 } x+\cfrac { 5 }{ 9 } } \right) +\cfrac { 44 }{ 27 } =\left( \cfrac { 3x+2 }{ 3 } \right) \left( 5x2-\cfrac { 11 }{ 3 } x+\cfrac { 5 }{ 9 } \right) +\cfrac { 44 }{ 27 } \)
= \(\left( \cfrac { 3x+2 }{ 3 } \right) 3\left( \cfrac { 5 }{ 3 } { x }^{ 2 }-\cfrac { 11 }{ 9 } x+\cfrac { 5 }{ 27 } \right) +\cfrac { 44 }{ 27 } \)
= \(\left( 3x+2 \right) \left( \cfrac { 5 }{ 3 } { x }^{ 2 }-\cfrac { 11 }{ 9 } x+\cfrac { 5 }{ 27 } \right) +\cfrac { 44 }{ 27 } \)
Hence the quotient \(\cfrac { 5 }{ 3 } { x }^{ 2 }-\cfrac { 11 }{ 9 } x+\cfrac { 5 }{ 21 } \) and remainder IS \(\cfrac { 44 }{ 27 } \)

17.

Given: AB = 2, BC = 6, AE = 6, BF = 8, CE = 7 and CF = 7
Consider ΔAEC and ΔBCF.
In ΔAEC, AE = 6, EC = 7 and AC = 8 (2 + 6 = 8)
In ΔBCF, BC = 6, CF = 1 and BF = 8
∴ ΔAEC ≅ BCF
∴ Area of ΔAEC = Area of ΔBCF (Two triangles are similar areas are equal)
Subtract area of ΔBDC on both sides we get,
Area of ΔAEC - Area of ΔBDC = Area of ΔBCF - Area of ΔBDC
Area of quadrilateral ABDE = Area of ΔCDF.
The answer is 1 : 1.

Generic 3

13 x 3 = 39

18.

Let E be the event that it will rain tomorrow. Then E′ is the event that it will not rain tomorrow.
Since P(E) = 0.91, we have P(E′) = 1−0.91 (how?)
= 0.09
Therefore, the probability that it will not rain tomorrow
= 0.09

19.

Given the dimensions of the cuboid;
that is length (l) = 7.5 m, breadth (b) = 3 m and height (h) = 5 m.
TSA = 2(lb + bh + lh)

= 2[(7.5 × 3) + (3 × 5) + (7.5 × 5)]
= 2(22.5 +15 + 37.5)
= 2 × 75
= 150 m²
LSA = 2(l + b) × h
= 2(7.5 + 3) × 5
= 2 ×10.5 × 5
= 105 m²

20.

(i) sin74° = sin(90⁰ -16⁰) (since, 90⁰ -16⁰ = 74⁰ )
RHS is of the form sin(90⁰ - \(\theta\)) = cos\(\theta\)
Therefore sin74° = cos16⁰
(ii) tan12° = tan(90⁰ - 78⁰) (since, 12⁰= 90⁰ = 78⁰ )
RHS is of the form tan(90⁰- \(\theta\)) = cot \(\theta\)
Therefore tan12° = cot 78⁰
(iii) cosec 39° = cosec(90⁰ - 51⁰) (since, 39⁰ = 90⁰ - 51⁰ )
RHS is of the form cosec(90⁰ - \(\theta\)) = sec\(\theta\)
Therefore cosec39° = sec51⁰

21.

Let p(x)= x³+ 9x²- x - 105
By factor theorem,x-3 is a factor of p(x),if p(3) = 0
p(3) = 33 + 9(3)³ - 3 - 105
= 27 + 81 - 3 -105
= 108-108
p(3) = 0
Therefore,x-3 is a factor of x³+ 9x²- x - 105

22.

\(\sqrt [ 5 ]{ 11 } +\sqrt [ 7 ]{ 11 } =\left( 5+7 \right) \sqrt { 11 } =12\sqrt { 11 } \)

23.

31, 33, 63, 33, 28, 29, 33, 27, 27, 34, 35, 28, 32
Writing in ascending order we get
27, 27, 28, 28, 29, 31, 32, 33, 33, 33, 34, 35, 36
Number of values = 13 which is an odd number
\(\therefore\) Median =\(\left( \cfrac{13+1}{2} \right) \) Value
= \(\left(\cfrac{14}{2}\right)^{th}\) Value = 7^th value
= 32

24.

(i) \( =\sqrt{8 \times 5}-\sqrt{4 \times 5} =\sqrt{4 \times 2 \times 5}-\sqrt{4 \times 5} \)
\( =2 \sqrt{2 \times 5}-2 \sqrt{5} \)
\( =2 \sqrt{2} \times \sqrt{5}-2 \sqrt{5} \)
\( =2 \times 2.236(1.414-1) \)
\( =4.472 \times 0.414 =1.852 \)
(ii) \(\sqrt{300}+\sqrt{90}-\sqrt{8}=\sqrt{3\times100}+\sqrt{9\times10}+\sqrt{4\times2}\)
\(=10\sqrt{3}+3\sqrt{10}+2\sqrt{2}\)
= 10\(\times\)1.732 + 3\(\times\)3.162 + 2\(\times\)1.414
= 17.32 + 9.486 + 2.828 = 23.978

25.

U = {x: −4 ≤ x ≤ 4, x \(\in \)Z}, U = {-4,-3,-2,-1,0,1,2,3,4}
A = {x:-4 B = {x :−2 ≤ x ≤ 3,x \(\in \) Z}, B = {-2,-1,0, 1,2,3}
(AUB)' = A'⋂B'
AUB = {-3, -2, -1,0, 1,2, 3}
(AUB)' = {-4,4} ...(1)
A' = {-4, 3, 4}; B' = {-4, -3, 4}
A'∩B' = {-4,4} ...(2)
From (1) and (2) it is verified that
(AUB)' = A'⋂B'

26.

(i) The algebraic expression is 3n + 1
(ii) The Algebraic expression is 3n - 2

27.

Let M be the set of students opting for Mathematics.
Let S be the set of students opting for Science.
n (M U S) = 40, n (M) = 26, n (S) = 24
n (M U S) = n (M) + n (S) - n (M n S)
40 = 26 + 24 - n (M n S)
n (M n S) = 26 + 24 - 40
= 50 - 40 = 10
∴ Number of students opted for Mathematics and Science = 10

28.

Joining the points we get the aeroplane shape.
(i) A (-2,0); F (10, 0) - x-axis
(ii) C(0, 2); I (0, -2) - y-axis
(iii) D(8,1); E(11,3) - I-Quadrant
B (-1, 1) - II - Quadrant
J (-1, -1) -III - Quadrant
G (11,-3); H (8, -1) - IV - Quadrant

29.

\( -5.132 =\frac{-5132}{1000} \\ =\frac{-1283}{250} \)

30.

Let the point O be (x, y), A be (3, 4) and B be (-5, 6).
Distance = \(\sqrt { ({ x }_{ 2 }-{ x }_{ 1 })^{ 2 }+(y_{ 2 }-y_{ 1 })^{ 2 } } \)
Given OA = OB
\(\sqrt { (x-3)^{ 2 }+(y-4)^{ 2 } } =\sqrt { (x+5)^{ 2 }+(y-6)^{ 2 } } \)
Squaring on both sides
(x-3)² + (y-4)² = (x+5)²+(y-6)²
x²-6x+9+y²-8y+16 = x²+10x+25+y²-12y+36
x²+y²-6x-8y+25 = x²+y²+10x-12y+61
-6x-10x-8y+12y = 61-25
⇒ -4x + y = 9
The relation between x and y is y = 4x + 9

Multiple Choice Question

14 x 1 = 14

31.

(b)
Fair

32.

(b)
6 cm²

33.

(a)
\(\frac { 1 }{ 2 } \)

34.

(b)
(2, 3)

35.

(b)
20%

36.

(a)
1

37.

(a)
\(3\sqrt [ 3 ]{ 2 } \)

38.

(b)
13

39.

(d)
12.9

40.

41.

(b)
10

42.

(d)
centroid

43.

44.

(b)
binomial

Generic 8

2 x 8 = 16

45.

Construction:
Step 1: Draw ΔABC with the given measurements AB = 8 cm, \(\angle\)A = 90^o and AC = 6 cm and construct the perpendicular bisector of any two sides (AB and AC) to find the mid points M and N of AB and BC respectively.
Step 2: Draw the medians (C and BN and let them meet at G. The point G is the centroid of the given ΔABC.

46.

In \(\triangle \)ABC, AB = 6 cm, LB= 110°, AC = 9 cm

Construction :
Step 1: Draw \(\triangle \)ABC with AB = 6 cm, ∠B =110°, AC = 9 cm
Step 2: Draw perpendicular bisectors of any two sides (BC and AB) to find the mid points of BC and AB.
Step 3: Construct medians AD and CE. Let them meet at G.
Step 4: G is the centroid of the given \(\triangle \)ABC.