Question

The two sides of a triangle are in the ratio of 2:3 and it's third side is 5cm . if the perimeter of the triangle is 15cm , what is it's area

Answer 1

$\large\underline{\sf{Solution-}}$

Let us consider a triangle ABC such that AB, BC and CA is represented by a, b, c respectively.

According to statement,

Let assume that,

• a : b = 2 : 3 and c = 5 cm.

Let a = 2x and b = 3x

Further given that, Perimeter of Triangle = 15 cm

⟼ a + b + c = 15

⟼ 2x + 3x + 5 = 15

⟼ 5x + 5 = 15

⟼ 5x = 15 - 5

⟼ 5x = 10

⇛ x = 2

Hence,

The sides of triangle ABC are

• ⟼ a = 2x = 2 × 2 = 4 cm

• ⟼ b = 3x = 3 × 2 = 6 cm

• ⟼ c = 5 cm

We know that,

$\underline{\boxed{\sf Semi \: Perimeter \ of \ a \ triangle \: (s)= \dfrac{1}{2} (a+b+c)}}$

$\rm :\longmapsto\:s = \dfrac{1}{2}(4 + 6 + 5)$

$\rm :\longmapsto\:s = \dfrac{1}{2}(15)$

$\bf\implies \:s = \dfrac{15}{2} \: cm$

We know,

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

$\rm \:= \: \: \sqrt{\dfrac{15}{2}\bigg(\dfrac{15}{2} - 4\bigg)\bigg(\dfrac{15}{2} - 6 \bigg)\bigg(\dfrac{15}{2} - 5\bigg) }$

$\rm \:= \: \: \sqrt{\dfrac{15}{2}\bigg(\dfrac{15 - 8}{2}\bigg)\bigg(\dfrac{15 - 12}{2} \bigg)\bigg(\dfrac{15 - 10}{2}\bigg) }$

$\rm \:= \: \: \sqrt{\dfrac{15}{2}\bigg(\dfrac{7}{2}\bigg)\bigg(\dfrac{3}{2} \bigg)\bigg(\dfrac{5}{2}\bigg) }$

$\rm \:= \: \:\dfrac{15}{4} \sqrt{7} \: {cm}^{2}$

Additional Information :-

$\green{\boxed{\bf{Area_{(rectangle)} = l \times b}}}$

$\green{\boxed{\bf{Area_{(square)} = 4 \times side}}}$

$\green{\boxed{\bf{Area_{(circle)} = \pi \: {r}^{2} }}}$

$\green{\boxed{\bf{Area_{(rhombus)} = h \times b}}}$

$\green{\boxed{\bf{Area_{(parallelogram)} = h \times b}}}$