Question

Sides of a triangle are in the ratio 12:17:25 and its perimeter in 540 cm. Find its area ​

Answer 1

$\huge\bf\underline\mathfrak{Answer :}$

• $\text {Area of the triangle = 9000 cm² ✓}$

$\huge\bf\underline\mathfrak{Concept :}$

• Application of Heron's formula when the sides are obtained.
• Measurement of the sides when the perimeter is given and then adding up the sides in their ratio form.

$\huge\bf\underline\mathfrak{Step \: by \: step \: explanation :}$

$\huge\bf\underline\mathfrak{Given :}$

• $\text {Perimeter of the triangle = 540 cm.}$
• $\text {Ratio of the sides = 12:17:25.}$

$\huge\bf\underline\mathfrak{To \: find :}$

• $\text {Area of the triangle.}$

$\huge\bf\underline\mathfrak{Solution :}$

Given, ratio of the sides = $\bf 12:17:25.$

∴ Let the sides be : $\bf 12x,17x, and 25x.$

• ( Where, x denotes any number. )

Let these sides denoted by ( a, b and c ) cm

Here, perimeter = 540 cm.

⇒ Semi-perimetre , s = Perimeter/2 = 540 / 2 = 270 cm.

Now, Perimeter = Sum of the sides

$\implies\bf\ { 540 = a + b + c }$

$\implies\bf\ { 540 = 17x + 12x + 25x }$

$\implies\bf\ { 540 = 54x }$

$\implies\bf\ x \: = \frac{540}{54}$

$\implies\bf\ {x = 10 \: cm }$

$\text {Now, measures of the sides :-}$

$\implies\bf\ { a = 12x = 12 × 10 = 120 \: cm }$

$\implies\bf\ { b = 17x = 17 × 10 = 170 \: cm }$

$\implies\bf\ { c = 25x = 25 × 10 = 250 \: cm }$

$\bf\underline { Area \: of \: the \: triangle :- }$

$\text { By Heron's formula :- }$

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

• Putting a = 120, b = 170, c = 250 and s = 270 cm in the above formula, we get :-

$\bf\sqrt{270(270 - 120)(270 - 170)(270 - 250)}$

$\implies\bf\sqrt{270 \times 150 \times 100 \times 120} \: cm ^{2}$

$\implies\bf\sqrt{(27 \times 15 \times 2) \times (10)^{5} }$

$\implies\bf\sqrt{(27 \times 3) \times (10)^{6} }$

$\implies\bf\sqrt{81 \times 10^{6} }$

$\implies\bf\sqrt{81 \times 1000000}$

$\implies\bf\sqrt{81000000}$

$\implies\bf\ (9 \times 1000 \: cm²)$

$\implies\boxed {9000 \: cm²}$ ★

Hence, area of the triangle is 9000 cm².