Question

3. Find the area of the triangle whose sides are 42 cm, 34 cm and 20 cm in<br />length. Hence, find the height corresponding to the longest side.​

Answer 1

$\huge\fcolorbox{red}{cyan}{16\:cm}$

Step-by-step explanation:

• Side 1 (a) = 42 cm
• Side 2 (b) = 34 cm
• Side 3 (c) = 20 cm

Now :-

$s = \frac{Perimeter}{2} \\ =>s = \frac{42+34+20}{2} \\ => s = \frac{96}{2} \\ =) \boxed{s = 48}$

Area of Triangle By Heron's Formula :-

$\bf\color{orange}{\sqrt{s(s-a)(s-b)(s-c)}} \\ =>\bf{Lets\:Put\:The\:Values} \\ =>\sqrt{48(48-42)(48-34)(48-20)} \\ =) \sqrt{48 \times 6 \times 14 \times 28} \\ => \sqrt{(6\times2\times4)\times6</p><p>\times(7\times2)\times(7\times4)} \\ => \sqrt{6</p><p>\times6\times2\times2\times4\times4\times7</p><p>\times7} \\ => 6 \times 2 \times 4\times7 \\ => \boxed{336 \: cm²}$

The Area Of Triangle Is 336 cm²

Now Lets Find The Height Corresponding To Longest Side :-

• Longest Side(base) = 42 cm
• Area = 336 cm²
• Height = ?

$\bf\color{purple}{Area\:Of\:Triangle = \frac{1}{2}\times base \times height} \\ => \frac{1}{2}\times 42 \times height = 336 \\ => 21\times height = 336 \\ => height = \frac{336}{21} \\ => \boxed{Height = 16 \: cm}$

Hope it helped

Answer 2

Given :-

First side = 42 cm

Second side = 34 cm

Third side = 20 cm

To Find :-

The area of the triangle.

The height corresponding to the longest side.

Solution :-

We know that,

• s = Semi perimeter
• a = Area
• h = Height
• b = Base

By the formula,

$\underline{\boxed{\sf Semi \ perimeter=\dfrac{a+b+c}{2} }}$

Given that,

First side = 42 cm

Second side = 34 cm

Third side = 20 cm

Substituting their values,

s = 42+34+20/2

s = 96/2

s = 48 cm

Therefore, the semi perimeter of the triangle is 48 cm.

Using Heron's formula,

$\underline{\boxed{\sf Area \ of \ triangle=\sqrt{s(s-a)(s-b)(s-c)} }}$

Given that,

Semi perimeter (s) = 48 cm

First side = 42 cm

Second side = 34 cm

Third side = 20 cm

Substituting their values,

$\sf =\sqrt{48(48-42) (48-34)(48-20) }$

$\sf =\sqrt{48 \times 6 \times 14 \times 28}$

$\sf =\sqrt{112896}$

$\sf =336 \ cm^2$

Therefore, the area of the triangle is 336 cm².

Let the height corresponding to longest side be 'x'.

By the formula,

$\underline{\boxed{\sf Area=\dfrac{1}{2} \times Base \times Height}}$

Substituting their values,

1/2 × 42 × x = 336

h = 336 × 2/42

h = 672/2

h = 16 cm

Therefore, the height corresponding to the longest side is 16 cm.