Question

The perimeter of a triangle is 60 cm. One side of a triangle is 6 cm longer than the smaller side and the third

side is 10 cm less than twice the smaller side. Find the area of the triangle.​

Answer 1

Step-by-step explanation:

Let the smaller side be = x cm Then, larger side = (x + 4) cm And, third side = (2x-6) cm Given that, Perimeter = 50 cm ⇒ x + (x + 4) + (2x-6) = 50 ⇒ 4x-2 = 50 ⇒ 4x = 52 ⇒ x = 13 Therefore, smaller side = 13cm Larger side = x + 4 = 13 + 4 = 17cm Third side = 2x-6 = 2×13 – 6 = 26-6 = 20cm To find area of triangle, Let a = 13, b = 17, c = 20 s = (a + b + c)/2 ⇒ s = (13 + 17 + 20)/2 = 50/2 = 25. Area of triangle = √s(s-a)(s-b)(s-c) = √25(25-13)(25-17)(25-20) = √25×12×8×5 = 20√30 cm2

Answer 2

Step-by-step explanation:

$\frak Given = \begin{cases} &\sf{The\ perimeter\ of\ the\ triangle\ is\ 60cm.} \\ &\sf{One\ side\ of\ a\ triangle\ is\ 6cm\ longer\ than\ the\ smaller\ side.} \\ &\sf{The\ third\ side\ of\ the\ triangle\ is\ 10cm\ less\ than\ the\ smaller\ side.} \end{cases}$

To find:- We have to find the area of the triangle ?

☯️ Let the side first side of the triangle be x cm. So, the second side be (x + 6). Then, the third side be (2x - 10).

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$\frak{\underline{\underline{\dag As\ we\ know\ that:-}}}$

$\sf : \implies {\gray{\underline{\boxed{\bf Area_{(triangle)}\ =\ \sqrt{s(s-a)(s-b)(s-c)}}}}}$

__________________

$\frak{\underline{\underline{\dag According\ to\ the\ question:-}}}$

Given:-

• First side = x cm.
• Second side = (x + 6)cm.
• Third side = (2x - 10)cm.
• Perimeter = 60cm.

Therefore:-

$\bf : \implies {x\ +\ (x\ +\ 6)\ +\ (2x\ -\ 10)\ =\ 60} \\ \\ \bf : \implies {4x\ =\ 64} \\ \\ \bf : \implies {x\ =\ \cancel \dfrac{64}{4}} \\ \\ \bf : \implies {\purple{\underline{\boxed{\bf x\ =\ 16cm.}}}}\bigstar$

Hence:-

• First side = x = 16cm.
• Second side = (x + 6) = 16 + 6 = 22cm.
• Third side = (2x - 10) = 2 × 16 - 10 = 22cm.

☯️ To find the area of the triangle we have to find the semi-perimeter of the triangle.

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$\sf \therefore {\gray{\underline{\boxed{\bf Semi-perimeter_{(triangle)}\ =\ \dfrac{16+22+22}{2}\ =\ 30cm.}}}}$

Now:-

$\frak{\underline{\underline{\dag By\ substituting\ the\ values\ in\ formula,\ we\ get:-}}}$

$\bf : \implies {\gray{\underline{\boxed{\bf Area\ =\ \sqrt{s(s-a)(s-b)(s-c)}}}}}$

Here:-

• s = 30cm.
• a = 16cm.
• b = 22cm.
• c = 22cm.

So now:-

$\bf : \implies {\sqrt{s(s-a)(s-b)(s-c)}} \\ \\ \bf : \implies {\sqrt{30(30-16)(30-22)(30-22)}} \\ \\ \bf : \implies {\sqrt{30×14×8×8}} \\ \\ \bf : \implies {\purple{\underline{\boxed{\bf 16\ \sqrt{105}\ cm^2}}}}\bigstar$

Hence:-

$\sf \therefore {\underline{The\ area\ of\ the\ triangle\ is\ 16\ \sqrt{105}\ cm^2}}$