PSEB Solutions for Class 9 Maths Chapter 12 Heron’s Formula
PSEB 9th Class Maths Solutions Chapter 12 Heron’s Formula
Ex 12.1
Question 1.
A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?
Answer:
In equilateral ∆ ABC, the length of each side is a.
Question 2.
The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see the given figure). The advertisements yield an earning of ₹ 5000 per m2 per year. A company hired one of its walls for 3 months. How much rent did it pay?
Question 3.
There is a slide in a park. One of its side walls has been painted in some colour with a message “KEEP THE PARK GREEN AND CLEAN” (see the given figure). If the sides of the wall are 15 m, 11m and 6 m, find the area painted in colour.
Question 4.
Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Answer:
Question 5.
Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area.
Answer:
Suppose the sides of the triangle measure 12x cm, 17x cm and 25x cm.
Perimeter of a triangle = Stun of three sides
∴ 540 = 12x + 17x + 25x
∴ 540 = 54x
∴ x = 10
Then, the measures of the sides of the triangle are,
a = 12 × 10 = 120 cm,
b = 17 × 10 = 170 cm and
c = 25 × 10 = 250 cm.
Now, s – a = 270 – 120 = 150 cm,
s – b = 270 – 170 = 100 cm and
s – c = 270 – 250 = 20 cm.
Area of a triangle
Question 6.
An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.
Answer:
Let, the sides of the isosceles triangle be a = 12 cm, b = 12 cm and c cm.
Perimeter of triangle = Sum of three sides
∴ 30 = 12 + 12 + c
∴ 30 = 24 + c
∴ c = 6 cm
Ex 12.2
Question 1.
A park, in the shape of a quadrilateral ABCD has ∠C = 90°, AB = 9 m, BC = 12 m, CD = 5 m and AD = 8 m. How much area does it occupy?
Answer:
In ∆ BCD, ∠C = 90°
∴ BD2 = BC2 + CD2
= (12)2 + (5)2
= 144 + 25
= 169
= (13)2
∴ BD = 13 m
Question 2.
Find the area of a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm.
Answer:
Question 3.
Radha made a picture of an aeroplane with coloured paper as shown in the given figure, s Find the total area of the paper used. ;
The right triangle in part V is congruent to the right triangle in part IV.
∴ Area of right triangle in part V = 4.5 cm2
Now, total area of the paper used
= Areas of figures in part I to part V
= (2.5 + 6.5 + 1.3 + 4.5 + 4.5) cm2
= 19.3 cm2
Question 4.
A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 26 cm, 28 cm and 30 cm, and the parallelogram stands on the base 28 cm, find the height of the parallelogram.
Question 5.
A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer diagonal is 48 m, how much area of grass field will each cow be getting?
Answer:
Question 6.
An umbrella is made by stitching 10 triangular pieces of cloth of two different colours (see the given figure), each piece measuring 20 cm, 50 cm and 50 cm. How much cloth of each colour is required for the umbrella?
Answer:
Out of 10 triangular pieces, 5 are dark coloured and 5 are light coloured.
For each triangle, a = 20 cm, b = 50 cm and c = 50 cm
Hence, the total area of 5 dark coloured cloth pieces = 5 × 200 √6 cm2 = 1000 √6 cm2
Similarly, the total area of 5 light coloured cloth pieces = 5 × 200 √6 cm2 = 1000 √6 cm2
Question 7.
A kite in the shape of a square with a diagonal 32 cm and an isosceles triangle of base 8 cm and sides 6 cm each is to be made of three different shades as shown in the given figure. How much paper of each shade has been used in it?
Answer:
Let us name the square part as ABCD and the triangular part as CMN.
Suppose the length of square ABCD is xcm.
∴ In ∆ ABD, AB = AD = x cm and ∠A = 90°
The length of hypotenuse BD is given to be 32 cm.
AB2 + AD2 = BD2 (Pythagoras’ theorem)
∴ (x)2 + (x)2 = (32)2
∴ 2x2 = 1024
∴ x2 = 512
∴ x = √512
Question 8.
A floral design on a floor is made up of 16 tiles which are triangular, the sides of the triangle being 9 cm, 28 cm and 35 cm (see the given figure). Find the cost of polishing the tiles at the rate of 50 p per cm2.
Answer:
For each of 16 triangular tiles,
a = 9 cm; b = 28 cm and c = 35 cm
= 88.2 cm2 (approx.)
∴ Area of 16 tiles = 16 × 88.2 cm2
= 1411.2 cm2
50 paise = ₹ 0.50
Cost of polishing 1 cm2 region = ₹ 0.50
∴ Cost of polishing 1411.2 cm2 region
= ₹ (1411.2 × 0.50)
= ₹ 705.60
Question 9.
A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The non-parallel sides are 14 m and 13 m. Find the area of the field.
Answer:
In the given figure, trapezium ABCD represents the field in which AB || CD,
AB = 25 m, BC = 14 m, CD = 10 m and DA = 13 m.
Through C, draw a line parallel to DA to intersect AB at E.
In quadrilateral AECD, AE || CD and DA || CE
∴ AECD is a parallelogram.
∴ CE = DA = 13 m and AE = CD = 10 m
Now, BE = AB – AE = 25 – 10 = 15 m
In ∆ CEB, a = 13 m; b = 15 m and c = 14 m
MCQ
Multiple Choice Questions and Answer
Answer each question by selecting the proper alternative from those given below each question to make the statement true:
Question 1.
The sides of a triangle measure 8cm, 12cm and 6 cm. Then, the semiperimeter of the triangle is ……………………… cm.
A. 26
B. 52
C. 13
D. 6.5
Answer:
C. 13
Question 2.
Each side of an equilateral triangle measures 8 cm. Then, the semiperimeter of the triangle is ……………………….. cm.
A. 4
B. 24
C. 12
D.36
Answer:
C. 12
Question 3.
In a right angled triangle, the length of the hypotenuse is 15 cm and one of the sides forming right angle is 9 cm. Then, the semiperimeter of the triangle is ……………………….. cm.
A. 36
B. 18
C. 12
D. 15
Answer:
B. 18
Question 4.
The ratio of the measures of the sides of a triangle is 3:4:5. If the semiperimeter of the < triangle is 36 cm, the measure of the longest side of the triangle is ……………………. cm.
A. 12
B. 15
C. 20
D. 30
Answer:
D. 30
Question 5.
The area of a triangle is 48 cm2 and one of its sides measures 12 cm. Then, the length of the altitude corresponding to this side is …………………. cm.
A. 4
B. 8
C. 16
D. 6
Answer:
B. 8
Question 6.
The sides of a triangle measure 12 cm, 17 cm and 25 cm. Then, the area of the triangle is ……………………….. cm2.
A. 54
B. 90
C. 180
D. 135
Answer:
B. 90
Question 7.
Two sides of a triangle measure 9 cm and 10 cm. If the perimeter of the triangle is 36cm, then its area is …………………. cm2.
A. 17
B. 36
C. 72
D. 18
Answer:
B. 36
Question 8.
The area of an equilateral triangle with each side measuring 10 cm is ………………….. cm2.
A. 5√3/2
B. 25√3
C. 5√3
D. 3√5
Answer:
B. 25√3
Question 9.
∆ ABC is an isosceles triangle in which BC = 8 cm and AB = AC = 5 cm. Then, area of ∆ ABC = ……………………….. cm2.
A. 6
B. 12
C. 18
D. 24
Answer:
B. 12
Question 10.
ABCD is a parallelogram. If ar (ABC) = 18 cm2, then ar(ABCD) = …………………. cm2.
A. 18
B. 9
C. 36
D. 27
Answer:
C. 36
Question 11.
ABCD is a parallelogram. If ar (ABC) = 18 cm2, then ar (ABCD) = …………………. cm2.
A. 3.6
B. 7.2
C. 7.5
D. 6
Answer:
B. 7.2
Question 12.
In quadrilateral ABCD, AC = 10 cm. BM and DN are altitudes on AC from B and D respectively. If BM = 12cm and DN = 4 cm, then ar (ABCD) = …………………. cm2.
A. 160
B. 80
C. 320
D. 480
Answer:
B. 80
Question 13.
The perimeter of rhombus ABCD is 40 cm and BD =16 cm. Then, ar (ABCD) = ……………………. cm2.
A. 96
B. 48
C. 24
D. 72
Answer:
A. 96
Question 14.
The area of a rhombus is 72 cm2 and one of its diagonals measures 16 cm. Then, the length of the other diagonal is ………………… cm.
A. 12
B. 9
C. 18
D. 15
Answer:
B. 9
Question 15.
PQRS is a square. If PQ = 10 cm, then PR = ……………………….. cm.
A. 10
B. 20
C. 10√2
D. 2√10
Answer:
C. 10√2