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Maths 3 Marks Question

Heron’s Formula · Gujarat Board (GSEB) STD 9 (GSEB - English Medium). 5 questions.

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Question 13 Marks
Sides of a triangle are in the ratio of 12: 17: 25 and its perimeter is 540 cm. Find its area.
Answer
Let the sides of the triangle be 12x,17x and 25x
Therefore, 12x+17x+25x = 540
$\Rightarrow$ 54x=540 $\Rightarrow$ x = 10
$\therefore$ The sides are 120 cm, 170 cm and 250 cm.
Semi-perimeter of triangle $s =\frac{120+170+250}{2}$ = 270 cm
Now, Area of triangle = $\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$
= $\sqrt{270\left( 270-120 \right)\left( 270-170 \right)\left( 270-250 \right)}$
= $\sqrt{270\times 150\times 100\times 20}$
= 9000cm2
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Question 23 Marks
The sides of a triangular plot are in the ratio of 3 : 5 : 7 and its perimeter is 300 m. Find its area.
Answer


Suppose that the sides in metres are 3x, 5x and 7x.
Then, we know that 3x + 5x + 7x = 300 (Perimeter of the triangle)
Therefore, 15x = 300, which gives x = 20.
So the sides of the triangles are 3 $\times$ 20 m, 5 $\times$ 20 m and 7 $\times$ 20 m
i.e., 60m, 100m and 140m.
We have s = $\frac{60+100+140}{2}$ = 150 m
and area will be = $\sqrt{150(150-60)(150-100)(150-140) }$
= $\sqrt{150 \times 90 \times 50 \times 10}$ 
= 1500$\sqrt{3}$ m2
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Question 33 Marks
A triangular park ABC has sides 120 m, 80 m and 50 m. (in a given figure). A gardener Dhania has to put a fence all around it and also plant grass inside. How much area does she need to plant? Find the cost of fencing it with barbed wire at the rate of ₹ 20 per metre leaving a space 3m wide for a gate on one side.
Answer
Computation of area: Clearly, the park is trianglar with sides 
a = BC = 120 m, b = CA = 80 m and c = AB = 50 m
Ifs denotes the semi-perimeter of the park, then
2s = a + b + c $\Rightarrow$ 2s = 120 + 80 + 50 $\Rightarrow$ s = 125
$\therefore$ s - a = 125 - 120 = 5, s - b = 125 - 80 = 45 and s - c = 125 - 50 = 75
Hence, Area of the park = $\sqrt{s(s-a)(s-b)(s-c)}$ = $\sqrt{125 \times 5 \times 45 \times 75}$m2 = 375$\sqrt{15}$ m2
Length of the wire needed for fencing = perimeter of the park - width of the gate
= 250m - 3m = 247 m
Cost of fencing = Rs.(20 $\times$ 247) = Rs.4940
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Question 43 Marks
Find the area of a triangle, two sides of which are 8 cm and 11 cm and the perimeter is 32 cm.
Answer


Let a, b, c be the sides of the given triangle and 2s be its perimeter such that
a = 8 cm, b = 11 cm and 2s = 32 cm i.e. s = 16 cm
Now,
a + b + c = 2s
$\Rightarrow$ 8 + 11 + c = 32
$\Rightarrow$ c = 13
$\therefore$ s - a = 16 - 8 = 8, s - b = 16 - 11 = 5 and s - c = 16 - 13 = 3
Hence, Area of given triangle = $\sqrt{s(s-a)(s-b)(s-c)}$ 
= $\sqrt{16 \times 8 \times 5 \times 3}$ = 8$\sqrt{30}$ cm2
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Question 53 Marks
Answer
1. 24 m²
2. 3696 m²
3. No, with valid reasoning
● No, the area reserved under part 1 is not equal to the area reserved under part 2. Area under part 1 is 3696 m² whereas the area under part 2 is 3024 m²
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