Question

12. If (s - a ) = 7 cm, (s - b ) = 8 cm and (s - c ) = 15 cm. Find s.
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Answer 1

Answer:

The semi perimeter ( s ) of the triangle is 30 cm.

Step-by-step-explanation:

We have given the subtraction of the sides of a triangle and the semi perimeter.

We have to find the semi perimeter of the triangle.

We have given that,

( s - a ) = 7 cm

( s - b ) = 8 cm

( s - c ) = 15 cm

Now,

( s - a ) + ( s - b ) + ( s - c ) = 7 + 8 + 15

⇒ s - a + s - b + s - c = 15 + 15

⇒ s - a + s - b + s - c = 30

⇒ s + s + s - a - b - c = 30

⇒ 3s = 30 + ( a + b + c )

⇒ ( 3s / 2 ) = ( 30 / 2 ) + ( a + b + c ) / 2 - - [ Dividing each term by 2 ]

⇒ 3s / 2 = 15 + s - - [ ∵ s = ( a + b + c ) / 2 ]

⇒ 3s = ( 15 + s ) * 2

⇒ 3s = 30 + 2s

⇒ 3s - 2s = 30

⇒ s = 30 cm

∴ The semi perimeter of the triangle is 30 cm.

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Verification:

We have,

s = 30 cm

( s - a ) = 7 cm

( s - b ) = 8 cm

( s - c ) = 15 cm

Now,

( s - a ) = 7

⇒ 30 - a = 7

⇒ 30 - 7 = a

⇒ a = 23

Now,

( s - b ) = 8

⇒ 30 - b = 8

⇒ 30 - 8 = b

⇒ b = 22

Now,

( s - c ) = 15

⇒ 30 - c = 15

⇒ 30 - 15 = c

⇒ c = 15

Now, we know that,

Semi perimeter of triangle = ( Sum of sides ) / 2

⇒ s = ( a + b + c ) / 2

⇒ 30 = ( 23 + 22 + 15 ) / 2

⇒ 30 = ( 45 + 15 ) / 2

⇒ 30 = 60 ÷ 2

⇒ 30 = 30

∴ LHS = RHS

Hence verified!

Answer 2

semi perimeter(s) of the triangle is 30

Question :

If (s - a) = 7 cm, (s - b) = 8 cm and (s - c) = 15cm. Find s.

Given :

(s - a ) = 7 cm,

( s - b ) = 8 cm,

(s - c ) = 15cm.

To Find :

Semi perimeter (s) of a triangle

solution :

We have the terms of side of triangle and the subtraction of semi perimeter.

Add the terms of semi perimeter and side of triangle,

$(s - a) + (s - b) + (s - c) = 7 + 8 + 15$

$s - a + s - b + s - c = 30$

$(s + s + s) + (- a - b - c) = 30$

$3s = 30+ (a + b + c)$

Divide the given terms by 2 on both sides,

$( \frac{3s}{2} ) = ( \frac{30}{2}) + (\frac{a + b + c}{2} )$

Since,

s is the semiperimeter of the triangle, that is,

$s = (\frac{a + b + c}{2} )$

substitute the value of s in equation,

$\frac{3s}{2} = 15 + s$

$3s = 2(15 + s)$

$3s = 30 + 2s$

$3 s- 2 s= 30$

$s = 30$

Verification :

Substitute the value of (s) in the terms,

$(s - a) = 7 \\ (30 - a) = 7 \\ - a = 7 - 30 \\ - a = - 23 \\ a = 23$

$(s - b) = 8 \\ (30 - b) = 8 \\ - b = 8 - 30 \\ - b = - 22 \\ b = 22$

$(s - c) = 15 \\ (30 - c) = 15 \\ - c = 15 - 30 \\ - c = - 15 \\ c = 15$

Now Substitute the values of sides of triangle in heron's formula which says ,

The sum of three sides of triangle divided by 2 is equal to semi perimeter of the triangle

$s = \frac{(a + b + c)}{2}$

$s = \frac{(23 + 22 + 15)}{2} \\s = \frac{60}{2} \\ s = 30$

Hence verified.