Question

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

2. Sides of a triangle are in the ratio or 12:17:25 and it's perimetre is 540 cm . Find it's area .

3. An isosceles triangle has perimetre 30 cm and each of the equal sides is 12 cm . find the area of the triangle ?

Explain with good explanation .


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Answer 1

Question-1:

Given:

• First side(a) = 18cm
• Second side(b) = 10cm
• Perimeter = 42cm

To find:

• Area of the triangle.

Solution:

We know that perimeter of the triangle is the sum of all sides of the triangle. By using perimeter of triangle formula, we are going to find the third side.

⟶ a+b+c = Perimeter

⟶ 18+10+c = 42

⟶ 28+c = 42

⟶ c = 42-28

⟶ c = 14cm

Hence, we have found the third side of the triangle. Now, let us find the semi-perimeter of the triangle. Here, let us denote semi-perimeter of the triangle as s.

⟶ s = Perimeter/2

⟶ s = 42/2

⟶ s = 21cm

To find the area of triangle, we are going to use Heron's formula:

⟶ Area∆ = $\rm{\sqrt{s(s-a)(s-b)(s-c)}}$

⟶ Area∆ = $\rm{\sqrt{21(21-18)(21-10)(21-14)}}$

⟶ Area∆ = $\rm{\sqrt{21(3)(11)(7)}}$

⟶ Area∆ = $\rm{\sqrt{21(21)(11)}}$

⟶ Area∆ = 21√11cm²

Hence, the area of triangle is 21√11cm².

Question-2:

Given:

• Sides of the Triangle are in the ratio of 12:17:25
• Perimeter of triangle = 540cm

To find:

• Area of the triangle

Solution:

Let the sides of the triangle be:

• a = 12x
• b = 17x
• c = 25x

We know that,

⟶ a+b+c = Perimeter

⟶ 12x+17x+25x = 540

⟶ 54x = 540

⟶ x = 540/54

⟶ x = 10cm

Hence, the sides are:

• a = 12x = 12×10 = 120cm
• b = 17x = 17×10 = 170cm
• c = 25x = 25×10 = 250cm

Now, let us find the semi-perimeter of the triangle. Here, let us denote semi-perimeter of the triangle as s.

⟶ s = Perimeter/2

⟶ s = 540/2

⟶ s = 270cm

By using Heron's formula, we get:

⟶ Area∆ = $\rm{\sqrt{s(s-a)(s-b)(s-c)}}$

⟶ Area∆ = $\rm{\sqrt{270(270-120)(270-170)(270-250)}}$

⟶ Area∆ = $\rm{\sqrt{270(150)(100)(20)}}$

⟶ Area∆ = $\rm{\sqrt{81,000,000}}$

⟶ Area∆ = 9000cm²

Hence, the area of triangle is 9000cm².

Question-3:

Given:

• Perimeter of isosceles triangle = 30cm
• First side(a) = 12cm
• Second side(b) = 12cm

To find:

• Area of the isosceles triangle.

Solution:

Let us find the third side of the isosceles triangle.

⟶ a+b+c = Perimeter

⟶ 12+12+c = 30cm

⟶ 24+c = 30cm

⟶ c = 30-24

⟶ c = 6cm

Now, let us find the semi perimeter of the triangle:

⟶ s = Perimeter/2

⟶ s = 30/2

⟶ s = 15cm

By using Heron's formula, we have:

⟶ Area∆ = $\rm{\sqrt{s(s-a)(s-b)(s-c)}}$

⟶ Area∆ = $\rm{\sqrt{15(15-12)(15-12)(15-6)}}$

⟶ Area∆ = $\rm{\sqrt{15(3)(3)(9)}}$

⟶ Area∆ = $\rm{\sqrt{15(9)(9)}}$

⟶ Area∆ = 9√15cm²

Hence, the area of the isosceles triangle is 9√15cm²

Answer 2

Answer:

1. 21√11 cm²
2. 9000 cm²
3. 9√15 cm²

Step-by-step explanation:

Question

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

Given

• First side (a) = 18 cm
• Second side (b) = 10 cm
• Perimeter = 42 cm

To find

• Area of triangle

Solution

↪ Perimeter of triangle = sum of all the sides of triangle

• 42 = 18 + 10 + third side
• 42 = 28 + third side
• third side = 42 - 28
• third side = 14

↪ Semi perimeter = perimeter/2

• 42/2
• 21

↪ Using Heron's formula :

• √{s(s-a)(s-b)(s-c)}

↪ Where :

• s - semi perimeter
• a - first side
• b - second side
• c - third side

↪ Substituting we get :

• √{21(21-18)(21-10)(21-14)}
• √4851
• 21√11

∴ The area is 21√11 cm².

Question

• Sides of a triangle are in the ratio or 12:17:25 and it's perimeter is 540 cm. Find it's area.

Given

• Sides of a triangle are in the ratio 12 : 17 : 25
• Perimeter of triangle = 540 cm

To find

• Area of triangle.

Solution

↪ Perimeter = 540 cm

↪ Let the sides be 12x , 17x, 25x where x is any number.

↪ Then,

• 12x + 17x + 25x = 540
• 54x = 540
• x = 10

↪ Sides are,

• 12x = 120 cm
• 17x = 170 cm
• 25x = 250 cm

↪ Semi perimeter = Perimeter/2

• 540/2
• 270 cm

↪ Area of triangle, by Heron's formula:

• √270(270 - 120)(270 - 170)(270 - 250)
• √270(150)(100)(20)
• √270(300000)
• √81000000
• 9000

∴ The area is 9000 cm².

Question

• An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

Given

• Perimeter of an isosceles triangle = 30 cm
• Each of the equal sides = 12 cm

To find

• Area of triangle.

Solution

↪ Each of the equal sides = 12 cm

• In an isosceles triangle 2 sides are equal in length, hence the two sides (a) and (b) are 12 cm.

↪ a + b + c = 30 cm (Perimeter)

• 12 + 12 + third side = 30
• 24 + c =30
• c = 6 cm

↪ Semi perimeter = Perimeter/2

• 30/2
• 15 cm

↪ Area of triangle (Heron's formula),

• √s(s - a)(s - b)(s - c)
• √15(15 - 12)(15 - 12)(15 - 6)
• √15(3)(3)(9)
• √15(81)
• √1215
• 9√5

∴ The area is 9√15 cm².