Question

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.
n2
2
1. The length of the sides of a triangle are in the
3:4:5 and its perimeter is 144 cm. The area o
triangle is
(a) 684 cm
(b) 664 cm
(c) 764 cm2
(d) 864 cm2
2. The area of an isosceles triangle, each of w
equal sides is 13 cm and whose base is 24 cm, i
(a) 60 cm
(b) 55 cm2​

Answer 1

The area of triangle is 864 cm² and the height corresponding to the longest side is 28.8 cm.

Step-by-step explanation:

The length of the sides of a triangle are in the ratio 3:4:5. Let the length of sides be 3x,4x,5x.

It is given that the perimeter of the triangle is 144 cm.

The value of x is 12. It means the length of sides are 36,48,60.

Using heron's formula the area of triangle is

Where,

The area of triangle is

The area of triangle is 864 cm².

The area of triangle is

The height corresponding to the longest side is 28.8 cm.

Answer 2

S O L U T I O N 1:

$\bf{\large{\underline{\bf{Given\::}}}}$

The length of the sides of a triangle are in the ratio 3:4:5 & it's perimeter is 144 cm.

$\bf{\large{\underline{\bf{To\:find\::}}}}$

The area of triangle.

$\bf{\large{\underline{\bf{Explanation\::}}}}$

Let the ratio of side of the triangle be r

A/q

$\longrightarrow\sf{3r+4r+5r=144}\\\\\longrightarrow\sf{12r=144}\\\\\longrightarrow\sf{r=\cancel{144/12}}\\\\\longrightarrow\bf{r=12\:cm}$

So;

• 1st side of triangle = 3r = 3(12) cm = 36 cm
• 2nd side of triangle = 4r = 4(12) cm = 48 cm
• 3rd side of triangle = 5r = 5(12) cm = 60 cm

$\boxed{\bf{Using\:Heron's\:formula\::}}}}}$

$\longrightarrow\sf{Semi-perimeter=\dfrac{Sum\:of\:all\:sides}{2} }\\\\\\\longrightarrow\sf{Semi-perimeter=\dfrac{a+b+c}{2} }\\\\\\\longrightarrow\sf{Semi-perimeter=\dfrac{36cm+48cm+60cm}{2}} \\\\\\\longrightarrow\sf{Semi-perimeter=\cancel{\dfrac{144}{2} }}\\\\\\\longrightarrow\bf{Semi-perimeter=72\:cm}}$

&

$\longrightarrow\sf{Area\:of\:\triangle =\sqrt{s(s-a)(s-b)(s-c)} }\\\\\longrightarrow\sf{Area\:of\:\triangle=\sqrt{72(72-36)(72-48)(72-60)} }\\\\\longrightarrow\sf{Area\:of\:\triangle=\sqrt{72(36)(24)(12)} }\\\\\longrightarrow\sf{Area\:of\:\triangle=\sqrt{746496} }\\\\\longrightarrow\bf{Area\:of\:\triangle=864\:cm^{2} }$

S O L U T I O N 2:

$\bf{\large{\underline{\bf{Given\::}}}}$

• 1st side of triangle = 13 cm
• 2nd side of triangle = 13 cm
• 3rd side of triangle = 24 cm

$\bf{\large{\underline{\bf{To\:find\::}}}}$

The area of triangle.

$\bf{\large{\underline{\bf{Explanation\::}}}}$

$\boxed{\bf{Using\:Heron's\:formula\::}}}}}$

$\longrightarrow\sf{Semi-perimeter=\dfrac{Sum\:of\:all\:sides}{2} }\\\\\\\longrightarrow\sf{Semi-perimeter=\dfrac{a+b+c}{2} }\\\\\\\longrightarrow\sf{Semi-perimeter=\dfrac{13cm+13cm+24cm}{2}} \\\\\\\longrightarrow\sf{Semi-perimeter=\cancel{\dfrac{50}{2} }}\\\\\\\longrightarrow\bf{Semi-perimeter=25\:cm}}$

&

$\longrightarrow\sf{Area\:of\:\triangle =\sqrt{s(s-a)(s-b)(s-c)} }\\\\\longrightarrow\sf{Area\:of\:\triangle=\sqrt{25(25-13)(25-13)(25-24)} }\\\\\longrightarrow\sf{Area\:of\:\triangle=\sqrt{25(12)(12)(1)} }\\\\\longrightarrow\sf{Area\:of\:\triangle=\sqrt{3600} }\\\\\longrightarrow\bf{Area\:of\:\triangle=60\:cm^{2} }$

Option (a).