Question

in herons formula (s-a)=5cm and (s-b)=10cm or (s-c)=1cm find a,b and c

Answer 1

we know that s = (a+b+c)/2

so, a + b + c - 2a = 10

    b + c - a = 10 -------------------------(1)

similarly,

a + c - b = 20 -----------------------(2)

and, a + b - c = 2 -----------------(3)

   

solving above Equations,

a = 11

b = 6

c = 15

Answer 2

We know that,

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

Given that,

$s-a = 5 \ \ \ \ \ \longrightarrow\ \ \ \ \ (1) \\ \\ \\ s-b=10 \ \ \ \ \ \longrightarrow\ \ \ \ \ (2) \\ \\ \\ s-c=1 \ \ \ \ \ \longrightarrow\ \ \ \ \ (3)$

$(1) + (2) + (3) \\ \\ \\ (s-a)+(s-b)+(s-c)=5+10+1 \\ \\ s-a+s-b+s-c=16 \\ \\ 3s-(a+b+c)=16 \\ \\ 3s-2s = 16 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [\because\ a+b+c=2s] \\ \\ s=16\ cm$

From (1),

$s-a=5 \\ \\ 16-a=5 \\ \\ a=16-5 \\ \\ a=\bold{11\ cm}$

From (2),

$s-b=10 \\ \\ 16-b=10 \\ \\ b=16-10 \\ \\ b=\bold{6\ cm}$

From (3),

$s-c=1 \\ \\ 16-c=1 \\ \\ c=16-1 \\ \\ c=\bold{15\ cm}$

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