AREA OF SCALENE TRIANGLE

Scalene triangle is a triangle with all sides of different lengths.

All angles are different, too.

So, no sides are equal and no angles are equal.

Formula for Area of Scalene Triangle :

= √[s(s - a)(s - b)(s - c)]

where

s = (a + b + c)/2

Here a, b and c are side lengths of the triangle. 

Example 1 :

Find the area of the scalene triangle whose length of sides are 12 cm, 18 cm and 20 cm.

Solution :

Because the lengths of the three sides are different, the triangle is scalene triangle. 

s = (a + b + c)/2

Substitute 12 for a, 18 for b and 20 for c.

 = (12 + 18 + 20)/2

= 50/2

= 25

Formula for area of scalene triangle :

= √[s(s - a)(s - b)(s - c)]

Substitute. 

= √[25(25 - 12)(25 - 18)(25 - 20)]

= √(25 x 13 x 7 x 5)

= 5√455 cm²

Example 2 :

The sides of a scalene triangle are 12 cm, 16 cm and 20 cm. Find the altitude to the longest side.

Solution :

In order to find the altitude to the longest side of a triangle, first we have to find the area of the triangle.

s = (a + b + c)/2

Substitute 12 for a, 16 for b and 20 for c.

s = (12 + 16 + 20)/2

= 48/2

= 24

Formula for area of scalene triangle :

= √[s(s - a)(s - b)(s - c)]

Substitute. 

= √[24 x (24 - 12) x (24 - 16) x (24 - 20)]

= √(24 x 12 x 8 x 4)

= 96 cm²

Because we want to find the altitude to the longest side, the longest side will be the base of the triangle as shown below.

Here, the longest side is 20 cm.

Area of the above triangle = 96 cm²

1/2 x 20 x h = 96

10h = 96

Divide each side by 10.

h = 9.6

So, the altitude to the longest side is 9.6 cm.

Example 3 :

The sides of a scalene triangle are in the ratio (1/2) : (1/3) : (1/4). If the perimeter is 52 cm, then find the length of the smallest side.

Solution :

From the given information, the sides the triangle are

x/2, x/3 and x/4

Perimeter of the triangle = 52 cm

x/2 + x/3 + x/4 = 52

(6x + 4x + 3x)/12 = 52

13x/12 = 52

13x = 624

x = 48

Lengths of the sides :

x/2 = 24

x/3 = 16

x/4 = 12

So, the length of smallest side is 12 cm.

Example 4 :

The area of the scalene triangle is 216 cm² and the sides are in the ratio 3 : 4 : 5. Find the perimeter of the triangle.

Solution :

From the given information, the sides the triangle are

3x, 4x and 5x

s = (3x + 4x + 5x)/2

s = 6x

Area of the triangle = 216 cm²

√[s(s - a)(s - b)(s - c)] = 216

√[6(6x - 3x)(6x - 4x)(6x - 5x)] = 216

√[6(3x)(2x(x] = 216

√(36x⁴) = 216

6x² = 216

x² = 36

x = 6

Lengths of the sides :

3x = 18

4x = 24

5x = 30

Perimeter of the given scalene triangle is

= 18 + 24 + 30

= 72 cm

Example 5 :

One side of a right angle scalene triangle is twice the other, and the hypotenuse is 10 cm. Find the area of the triangle.

Solution :

Let 'x' be the length of one of the legs of the triangle.

Then, the length of the other leg is 2x. 

Using Pythagorean theorem,

x² + (2x)² = 10²

x² + 4x² = 100

5x² = 100

x² = 20

√x² = √20

x = √(4 x 5)

x = 2√5

Then,

2x = 2(2√5)

= 4√5

Area of the given right scalene triangle is

= (1/2)(x)(2x)

= (1/2)(2√5)(4√5)

= 20 cm²

Example 6 :

The area of the triangle is 21 cm² Calculate y, the length of the base.

Solution :

Area of traingle = (1/2) x base x height

(1/2) x y x 6 = 21

3y = 21

y = 21/3

y = 7 cm

So, the base of the triangle is 7 cm.

Below is a right-angled triangle and a rectangle.

Example 7 :

The area of the right-angled triangle is equal to the area of the rectangle. Calculate x

Solution :

Area of triangle = area of rectangle

(1/2) x base x height = length x width

(1/2) ⋅ 5 ⋅ 12 = 10 ⋅ x

30 = 10x

x = 30/10

x = 3 cm

So, width of the rectangle is 3 cm.

Example 8 :

Below is a diagram of a right-angled triangle and a square.

The area of the square is twice the area of the triangle. Calculate the length of each side of the square.

Solution :

Area of square = 2 x (1/2) x base x height

Let a be the side length of square

base of triangle = 16 cm and height = 4 cm

a2 = 16 x 4

a = √16 x 4

a = 4 x 2

a = 8 cm

So, side length of the square is 8 cm.

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