RD Sharma Solutions Ex-12.2, (Part -2), Heron's Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics PDF Download

Q8. Find the area of the quadrilateral ABCD in which AD = 24 cm, angle BAD = 90^∘ and BCD forms an equilateral  triangle whose each side is equal to 26 cm. [Take √3 = 1.73]

Solution:

Given that, in a quadrilateral ABCD in which AD = 24 cm,

Angle BAD = 90∘

BCD is an equilateral triangle and the sides BC = CD = BD = 26 cm

In triangle BAD, by applying Pythagoras theorem,

BA² = BD²−AD²

BA² = 26²+24²

Area of the triangle

Area of the triangle

Area of the triangle BAD = 120 cm²

Area of the equilateral triangle

Area of the equilateral triangle QRS 

Area of the equilateral triangle BCD = 292.37 cm²

Therefore, the area of quadrilateral ABCD = area of triangle BAD + area of the triangle BCD

The area of quadrilateral ABCD = 120 + 292.37

= 412.37 cm²

Q9.  Find the area of quadrilateral ABCD in which AB = 42 cm, BC = 21 cm, CD = 29 cm, DA = 34 cm and the diagonal BD = 20 cm.

Solution:

Given

AB = 42 cm, BC = 21 cm, CD = 29 cm, DA = 34 cm, and the diagonal

BD = 20 cm.

Now, for the area of triangle ABD

Perimeter of triangle ABD 2s = AB + BD + DA

2s = 34 cm + 42 cm + 20 cm

s = 48 cm/p>

By using Heron’s Formula,

Area of the triangle 

= 336cm²

Now, for the area of triangle BCD

Perimeter of triangle BCD 2s = BC + CD + BD

2s = 29cm + 21cm + 20cm

s = 35 cm

By using Heron’s Formula,

Area of the triangle BCD

= 210cm²

Therefore, Area of quadrilateral ABCD = Area of triangle ABD + Area of triangle BCD

Area of quadrilateral ABCD = 336 + 210

Area of quadrilateral ABCD = 546 cm²

Q10. Find the perimeter and the area of the quadrilateral ABCD in which AB = 17 cm, AD = 9 cm, CD = 12 cm, AC = 15 cm and angle ACB = 90^∘.

Solution:

Given are the sides of the quadrilateral ABCD in which

AB = 17 cm, AD = 9 cm, CD = 12 cm, AC = 15 cm and an angle ACB = 90^∘

By using Pythagoras theorem

BC² = AB²−AC²

BC² = 17²−15²

BC = 8 cm

Now, area of triangle

area of triangle 

area of triangle ABC = 60 cm²

Now, for the area of triangle ACD

Perimeter of triangle ACD 2s = AC + CD + AD

2s = 15 + 12 +9

s = 18 cm

By using Heron’s Formula,

Area of the triangle ACD

= 54cm²

Area of quadrilateral ABCD = Area of triangle ABC + Area of triangle ACD

Area of quadrilateral ABCD = 60 cm² + 54cm²

Area of quadrilateral ABCD = 114 cm²

Q11. The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm, and the diagonal AC measures 42 cm. Find the area of parallelogram.

Solution:

Given,

The adjacent sides of a parallelogram ABCD measures 34 cm and 20 cm, and the diagonal AC measures 42 cm.

Area of the parallelogram = Area of triangle ADC + Area of triangle ABC

Note: Diagonal of a parallelogram divides into two congruent triangles

Therefore,

Area of the parallelogram = 2 × (Area of triangle ABC)

Now, for area of triangle ABC

Perimeter = 2s = AB + BC + CA

2s = 34 cm + 20 cm + 42 cm

s = 48 cm

By using Heron’s Formula,

Area of the triangle ABC = 

= 336cm²

Therefore, area of parallelogram ABCD = 2 × (Area of triangle ABC)

Area of parallelogram = 2 × 336cm²

Area of parallelogram ABCD = 672 cm²

Q12. Find the area of the blades of the magnetic compass shown in figure given below.

Solution:

Area of the blades of magnetic compass = Area of triangle ADB + Area of triangle CDB

Now, for the area of triangle ADB

Perimeter = 2s = AD + DB + BA

2s = 5 cm + 1 cm + 5 cm

s = 5.5 cm

By using Heron’s Formula,

Area of the triangle 

= 2.49cm²

Also, area of triangle ADB = Area of triangle CDB

Therefore area of the blades of the magnetic compass = 2  × area of triangle ADB

Area of the blades of the magnetic compass = 2 ×2.49

Area of the blades of the magnetic compass = 4.98 cm²

Q13. A hand fan is made by sticking 10 equal size triangular strips of two different types of paper as shown in the figure. The dimensions of equal strips are 25 cm, 25 cm and 14 cm. Find the area of each type of paper needed to make the hand fan.

Solution:

Given that,

The sides of AOB

AO = 25 cm

OB = 25 cm

BA = 14 cm

Area of each strip = Area of triangle AOB

Now, for the area of triangle AOB

Perimeter = AO + OB + BA

2s = 25 cm +25 cm + 14 cm

s = 32 cm

By using Heron’s Formula,

Area of the triangle 

= 168cm²

Also, area of each type of paper needed to make a fan = 5 ×  Area of triangle AOB

Area of each type of paper needed to make a fan = 5 × 168 cm²

Area of each type of paper needed to make a fan = 840cm²

Q14. A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 13 cm, 14 cm and 15 cm and the parallelogram stands on the base 14 cm, find the height of a parallelogram.

Solution:

The sides of the triangle DCE are

DC = 15 cm,

CE = 13 cm,

ED = 14 cm

Let the h be the height of parallelogram ABCD

Now, for the area of triangle DCE

Perimeter = DC + CE + ED

2s  = 15 cm + 13 cm + 14 cm

s = 21 cm

By using Heron’s Formula,

Area of the triangle 

= 84cm²

Also, area of triangle DCE = Area of parallelogram ABCD ⇒ 84cm²

24 × h  = 84cm²

h = 6 cm.