4 Marks Questions

Question 14 Marks

Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.

Answer

a = 18 cm, b = 10 cm.
Perimeter = 42 cm.
⇒ a + b + c = 42
∴ 18 + 10 + c = 42
∴ 28 + c = 42
∴ c = 42 – 28
∴ c = 14 cm.
s = $\frac{42}{2}$ = 21 cm
∴ Areaṁ of the triangle $=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{21(21-18)(21-10)(21-14)}$
$=\sqrt{21(3)(11)(7)}$
$=\sqrt{(7)(3)(3)(11)(7)}$
= (7)(3)$\sqrt{11}$
= 21$\sqrt{11}$cm²

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Question 24 Marks

There is 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 figure). If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.

Answer

Since, sides of coloured triangular wall are 15 m, 11 m and 6 m.
$\therefore$ Semi-perimeter of coloured triangular wall
S =$\frac{15+11+6}{2}$=$\frac{32}{2}$=16 m
Now, Using Heron’s formula,
Area of coloured triangular wall
=$\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$
=$\sqrt{16\left( 16-15 \right)\left( 16-11 \right)\left( 16-6 \right)}$
=$\sqrt{16\times 1\times 5\times 10}$
=$20\sqrt{2}{{m}^{2}}$
Hence area painted in blue colour = $20\sqrt{2}{{m}^{2}}$

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Question 34 Marks

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 Fig.). 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?

Answer

Given: = 122 m, = 22 m and = 120 m
Semi-perimeter of triangle (s)$ =\frac{122+22+120}{2}=\frac{264}{2}$ = 132 m Using Heron’s Formula,
Area of triangle =$ \sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$
=$\sqrt{132\left( 132-122 \right)\left( 132-22 \right)\left( 132-120 \right)}$
$= \sqrt {132 \times 10 \times 110 \times 12} $

$ = \sqrt {11 \times 12 \times 10 \times 10 \times 11 \times 12} $
= 10 $\times$ 11 $\times$ 12
= 1320 m²
$\because $ Rent for advertisement on wall for 1 year = Rs. 5000 per$ {{m}^{2}}$
$\therefore $ Rent for advertisement on wall for 3 months for 1320 m²; $\frac{5000}{12}\times 3\times 1320$
= Rs.1650000
Hence rent paid by company = Rs. 16,50,000

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Question 44 Marks

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

A traffic signal board is an equilateral triangle with side a.
Perimeter of the signal board,
2s = a + a + a
$\Rightarrow \mathrm{s}=\frac{3}{2} a$
Area of triangle $=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{\frac{3 a}{2}\left(\frac{3}{2} a-a\right)\left(\frac{3}{2} a-a\right)\left(\frac{3}{2} a-a\right)}$
$=\sqrt{\frac{3 a}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}}=\sqrt{\frac{3 a^{4}}{16}}=\frac{\sqrt{3}}{4} a^{2}$ sq. units
Now, if perimeter = 180 cm
3a = 180
$\Rightarrow$ a = 60 cm
$\therefore \text { Area of signal board }= \frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times(60)^{2}=900 \sqrt{3} \mathrm{cm}^{2}$
So, area of the signal board is $900 \sqrt{3} \mathrm{cm}^{2}$.

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