Question

The perimeter of a triangular ground is 900 & it's sides are in the ratio 3:5:6. Using Heron's formula find the area of the ground. ​

Answer 1

Answer:

33750 m².

Step-by-step explanation:

Let sides of the triangle are 3X , 5X and 4X.

• Perimeter of triangle = 900 m
• Sum of all three sides = 900 m

• 3X + 5X + 4X = 900
• 12X = 900
• X = (900/12) m
• X = 75 m

Therefore,

• 3X = 3 × 75 = 225 m
• 5X = 5 × 75 = 375 m

And,

• 4X = 4 × 75 = 300
• Sides of the triangle are 225 m , 375 m and 300 m.

Let,

• A = 225 m
• B = 375 m

And,

• C = 300 m

Semi perimeter ( S ) = 1/2 × ( A + B + C )

• => 1/2 × ( 225 + 375 + 300 )
• => 1/2 × ( 900 )
• => 450 m

( S - A ) = 450 - 225 = 225 m

( S - B ) = 450 - 375 = 75 m

And,

( S - C ) = 450 - 300 = 150 m

Therefore,

Area of triangle = ✓S ( S - A ) ( S - B ) ( S - C )

• => ✓450 × 225 × 75 × 150
• => ✓1139062500
• => 33750 m².

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Answer 2

Step-by-step explanation:

$\sf \large \green{ \frak{Correct \: Question:- }}$

• The perimeter of a triangular ground is 900 cm & it's sides are in the ratio 4:5:6. Using Heron's formula. Find the area of the ground.

$\sf \large\green{ \frak{Given:- } }$

• The perimeter of a triangular ground is 900 cm and it's sides are in the ratio 3:5:6.

$\sf \large \blue{ \frak{To \: Find : - }}$

• The area of the ground.

$\sf \green {\large{ \frak{Solution:-}}}$

✇ Let us assume that the three sides of the ∆ be 4x,5x and 6x

$\sf \clubs \: As \: we \: know \: that -$

$\mapsto \boxed{ \color{violet} \bf\bigstar \: Perimeter \: of \: \triangle = Sum \: of \: all \: sides}$

So,

ACQ,

$\therefore \: \tt 4x + 5x + 6x = 900 \\ ☞ \tt \: 15x = 900 \\ ☞ \tt \: x = \cancel \frac{900}{15} \\ ☞\tt \large \ \pink{\: \boxed{ \frak{x = 60}}}$

So,

$➳ \sf \: First \: side = 4x \\ = \sf \: 4 \times 60 \\ = \bf \: 240 \: m \\ \\ \sf ➳ \: Second \: side = 5x \\ = \sf \: 5 \times 60 \\ = \bf \: 300 \: m \\ \\ ➳ \sf \: Third \: side = 6x \\ = \sf 6 \times 60 \\ \bf \: = \: 360 \: m$

★ Now to find Area,

$\mapsto \color{violet} \boxed{ \bf \bigstar \: Area \: of \: a \: \triangle = \sqrt{s(s - a)(s - b)(s - c}}$

So,

$\sf \: s = \frac{a + b + c}{2} \\ \\ ➟ \sf \: s = \frac{240 + 300 + 360}{2} \\ \\ \sf \: ➟ \: s = \frac{900}{2} \\ \\ \bf \:➟ \: \red{s = 450}$

$\sf \: Area _{(Triagle)} = \sqrt{s(s - a)(s - b)(s - c)} \\ \\ \sf Area _{(Triagle)} = \sqrt{450(450 - 240)(450 - 300)(450 - 360)} \\ \\ \sf Area _{(Triagle)} = \sqrt{450 \times210 \times 150 \times 90} \\ \\ \sf \: Area _{(Triagle)} = \sqrt{1275750000} \\ \\ \color{indigo}\bf \boxed{ \Large{ \frak{Area _{(Triagle)} = 35717.64 cm²(approx.}}}$

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