Question

The perimeter of an isosceles triangle is 44 cm and ratio of the equal side to its base is 4 : 3. The area of the triangle is:​

Answer 1

✯ ᴀɴsᴡᴇʀ ✯

★ given →

☞︎︎︎ the perimeter of triangle is 44cm and ratio of equal side to its base is 4:3

★ to find →

☞︎︎︎ area of the triangle?

★ solution →

☞︎︎︎ we have ratio of equal side to base is 4:3

let equal side is 4x and base is 3x

then, three sides of triangle are;

• 4x , 4x and 3x

we know that,

sum of sides of triangle = perimeter of triangle

$\bold{➪ \: 4x + 4x + 3x = 44 } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{➪ \: 11x = 44} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{➪ \: x = \cancel{\frac{44}{11} }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{➪ \: x = 4} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

so, common multiple of sides is x = 4

so that, three sides of triangle of triangle are

• 4x = 4 × 4 = 16cm
• 4x = 4 × 4 = 16cm
• 3x = 3 × 4 = 12cm

_________________________

✩ now find the area of triangle by using heron's formula

→ if a,b and c are sides of triangle then,

$\orange{ \bold { area \: = \sqrt{s(s - a)(s - b)(s - c)} }} \: \: \\ \\ \orange{ \bold{where, \: s = \frac{a + b + c}{2} }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

_________________________

let here,

a = 16cm , b = 16cm and c = 12cm

then,

$\bold{s = \frac{a + b + c}{2} } \\ \\ { \tt: \implies \: s = \frac{16 + 16 + 12}{2} } \\ \\ { \tt : \implies \: s = \frac{44}{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{ : \implies \: s = 22 } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

then, area of this triangle(A) ;

${ \tt A = \sqrt{s(s - a)(s - b)(s - c)} } \: \: \: \: \: \: \: \: \: \\ \\ { \tt :➪ \: A = \sqrt{22(22 - 16)(22 - 16)(22 - 12)} } \\ \\ { \tt : ➪ \: A = \sqrt{22 \times 6 \times 6 \times 10} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ { \tt: ➪ \: A = \sqrt{2 \times 11 \times 6 \times 6 \times 2 \times 5} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ { \tt: ➪ \: A = 2 \times 6 \sqrt{11 \times 5} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \pink{\bold{: ➪ \: A = 12 \sqrt{55} }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

hence, area of triangle is 12√55 cm² or 88.99cm²(approx)

Answer 2

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Answer:-

• $\underline{\boxed{\large\rm{Area = 88.99\:cm^{2} }}}$

Given:-

• Perimeter of the isosceles triangle = 44 cm.

• Ratio of it's side and base = 4:3 = 4/3

To Find:-

• Area of the triangle.

Solution:-

An isosceles triangle has two equal sides.

So,

Let the two equal side be 4x.

Let the base be 3x.

Then,

$\sf\underline{\boxed{\large{\mathfrak{perimeter = side + base}}}}$

$\rm { : \implies \: \: 4x + 4x + 3 x=44}$

$\rm { : \implies \: \: 8x+3x=44}$

$\rm { : \implies \: \: 11x=44}$

$\rm { : \implies \: \: x= \dfrac{44}{11} }$

$\rm:\implies \underline{\boxed{\pink{\mathfrak{x = 4}}}}\:\:\star$

Therefore,

➼ Side (a) = 4 × 4 = 16 cm

➼ Side (b) = 4 × 4 = 16 cm

➼ Base (c) = 3 × 4 = 12

Now,

$\rm { : \implies \: \: s=\dfrac{a+b+c}{2} }$

$\rm { : \implies \: \: s=\dfrac{16+16+12}{2} }$

$\rm { : \implies \: \: s=\dfrac{32+12}{2} }$

$\rm { : \implies \: \: s=\dfrac{44}{2} }$

$\rm:\implies \underline{\boxed{\blue{\mathfrak{s=22}}}}\:\:\star$

Formula to find the area :-

$\sf\underline{\boxed{\large{\mathfrak{ \sqrt{s(s-a)(s-b)(s-c)} }}}}$

$\rm { : \implies \: \: \sqrt{22(22-16)(22-16)(22-12)} }$

$\rm { : \implies \: \: \sqrt{22 \times(6)\times(6)\times(10)} }$

$\rm { : \implies \: \: \sqrt{2\times11\times6\times6\times2\times5} }$

$\rm { : \implies \: \: 2 \times6 \sqrt{11\times5} }$

$\rm { : \implies \: \: 12\sqrt{55} }$

$\rm:\implies \underline{\boxed{\purple{\mathfrak{88.99\:cm^{2} }}}}\:\:\bigstar$

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