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Given below are the **Class 9 Maths** Important Questions for Heron Formula Mensuration

b) Calculation problems

c) Multiple choice questions

d) Long answer questions

e) Fill in the blank's

1. Calculate the area in each case

- Triangle have sides as a=5 cm ,b=4 cm,c=3 cm
- Equilateral triangle having side a=2 cm
- Right angle triangle have base=4 cm and Height =3 cm
- Square whose diagonal is 10 cm
- Rectangle whose length and breath are 6 and 4 cm
- Parallelogram whose two sides are 10 cm and 16 cm and diagonal is 14 cm
- Parallelogram whose base is 10 cm and height is 14 cm
- Rhombus of diagonals to 10 and 24 cm
- Two sides of trapezium are 36 and 24 cm and its altitude is 12 cm.

**Solution **

Area

$A=\sqrt{s(s-a)(s-b)(s-c)}$=6cm

b) Area of equilateral

$A=\frac{\sqrt{3}}{4}a^{2}=\sqrt{3}$

c) Area of triangle

A=(1/2)BH=6cm^{2}

d) Area of square in terms diagonal

A=(1/2)d^{2}=50cm^{2}

e) Rectangle area is given by

A=LXB=24cm^{2}

f) In parallelogram whose two sides and diagonal are given, Area is given by

$A=\sqrt{s(s-a)(s-b)(s-d)}$

Where $s=\frac{a+b+d}{2}$

So s=20cm

So A=80(3)^{1/2}cm^{2}

g) Area is given by

A=Base X height =10X14=140cm^{2}

h) Area is given by

A=(1/2)d_{1}d_{2}=120cm^{2}

i) Area of trapezium is given by

A=(1/2)(Sum of parallel sides) Altitude

A=360cm^{2}

2) **True or False statement**

a) Heron formula for area of triangle is not valid of all triangles

b) If each side of the triangles is tripled, the area will becomes 9 times

c) Base and corresponding altitude of the parallelogram are 8 and 5 cm respectively.Area of parallelogram is 40 cm^{2}

d) If each side of triangle is doubled, the perimeter will become 4 times

e) If p is the perimeter of the triangle of sides a,b,c ,the area of triangle is

$A=\frac{1}{4}\sqrt{p(p-2a)(p-2b)(p-2c)}$

f) When two triangles are congruent, there areas are same

g) Heron’s belongs to America

h) If the side of the equilateral triangle is a rational number, the area would always be irrational number

**Solution **

- False
- True
- True
- False
- True
- True
- False
- True

__Multiple choice Questions__

3) The difference between sides at right angles in a right angled triangle is 14 cm. The area of the triangle is 120 cm^{2} . The perimeter of the triangle is

a) 80

b) 45

c) 60

d) 64

**Solution** (c)

Let y be one of the at right angle ,then another side will be y-14

Now we know that

A=(1/2)BH

120=(1/2)y(y-14)

y^{2}-14y-240

(y-24)(y+10)=0

y=24

So other side is 10

From pythogrous theorem

$hyp=\sqrt{10^{2}+24^{2}}$=26cm

So perimeter will be =10+24+26=60 cm

4. Find the area of the equilateral triangle whose perimeter is 180 cm?

Which of the following is true?

a)900(3)^{1/2} cm^{2}

b)300(3)^{1/2} cm^{2}

c)200(3)^{1/2} cm^{2}

d) None of these

**Solution **

P=3a => a=P/3=60 cm

Area of equilateral

$A=\frac{\sqrt{3}}{4}a^{2}$

5) One diagonal and side of the rhombus are 24 and 13 cm respectively, Find the Area and other diagonal size?

a) 160 cm^{2},10 cm

b) 120 cm^{2},20 cm

c) 120 cm^{2},10 cm

d) None of the above

**Solution** (c)

6) ABCD is a trapezium with AB =10cm, AD=5 cm, BC=4 cm and DC =7 cm?

Find the area of the ABCD

a) 34 cm^{2}

b) 28cm^{2}

c) 20 cm^{2}

d) None of these

**Solution** a

BC is the altitude between the two parallel sides AB and DC

So Area of trapezium will be given by

$A=\frac{1}{2}BC(AB+DC)$=34cm^{2}

7) Find the area and perimeter of the right angle triangle whose hypotenuse is 5 cm and Base is 4 cm

a) 6 cm^{2} ,12 cm

b) 12 cm^{2} ,14 cm

c) 4 cm^{2}, 6 cm

d) 12 cm^{2} ,6 cm

**Solution** (a)

By pythogorous theorem

$height=\sqrt{hyp^{2}-base^{2}}=\sqrt{25-19}=3$

So Area =(1/2) XBase X height=6 cm^{2}

Perimeter = 5+4+3=12 cm

8) In an isosceles triangle ?ABC with AB = AC=13 cm. D is mid point on BC. Also BC=10 cm

Which of the following is true?

a) Area of Triangle ABD and ADC are equal

b) Area of triangle ABD is 30 cm^{2}

c) Area of triangle ABC is 60 cm^{2}

d) All the above

**Solution** (d)

ABD an ADC are congruent triangle, So Area of Triangle ABD and ADC are equal

Also From pythogorous theorem, AD will be given as

$AD=\sqrt{AB^{2}-BD^{2}}=\sqrt{169-25}=12cm$

So Area of triangle ABC=(1/2)X base X height=60 cm^{2}

9) A triangle and a parallelogram have the same base and the same area. The sides of the triangle are 26 cm and 30 cm and parallelogram stands on the base 28 cm. calculate the height of the parallelogram

a) 12 cm

b) 14 cm

c) 10cm

d) 13 cm

**Solution** (a)

For triangle, all the sides are given, calculating the area using Heron formula

A=336 cm^{2}

Now for parallelogram, Area is given by

A= Base X Altitude

336=28 X H

Or H=12 cm

10) Find the percentage increase in size of the triangle if each side is doubled

a)

b) 300%

c) 400%

d) None of the above

**Solution** (a)

__Match the column__

Perimeter of rectangle of length 24 cm and diagonal 26 cm |
22cm |

Perimeter of the square of side 10 cm |
17 cm |

Triangle of sides 4,5,6 cm respectively |
40 cm |

Perimeter of parallelogram of two sides 5 and 6 cm respectively |
68 cm |

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