Question

In ∆ABC, If r1=8, r2=12, r3=24,Show that a=12,b=16,c=20. (From Properties of Triangles)​

Answer 1

Step-by-step explanation:

Given, r₁ = 8, r₂ = 12, r₃ = 24.

∴ Derivative of (1/r) = (1/r₁) + (1/r₂) + (1/r₃)

⇒ (1/r) = (1/8) + (1/12) + (1/24)

⇒ (1/r) = (3 + 2 + 1)/24

⇒ r = 4.

Now,

∴ Δ = √rr₁r₂r₃

⇒ Δ = √4 * 8 * 12 * 24

       = 96

(i)

r = Δ/s

⇒ 4 = 96/s

⇒ 4s = 96

⇒ s = 24.

(ii)

r₁ = Δ/s - a

⇒ 8 = 96/24 - a

⇒ 192 - 8a = 96

⇒ a = 12

(iii)

r₂ = Δ/s - b

⇒ 12 = 96/24 - b

⇒ 288 - 12b = 96

⇒ b = 16

(iv)

r₃ = Δ/s - c

⇒ 24 = 96/24 - c

⇒ 576 - 24c = 96

⇒ 480 = 24c

⇒ c = 20.

Therefore, a = 12, b = 16, c = 20.

Hope it helps!

Answer 2

$\huge\bold{Solution}$

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Given:- r1 = 8 , r2 = 12 , r3 = 24

Show that:- a = 12 , b = 16 , c = 24

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We know that ,

$\dfrac{1}{r}$ = $\dfrac{1}{r1}+\dfrac{1}{r2}+\dfrac{1}{r3}$

$\dfrac{1}{r}$ = $\dfrac{1}{8}+\dfrac{1}{12}+\dfrac{1}{24}$

$\dfrac{1}{r}$ = $\dfrac{3+2+1}{24}$

$\dfrac{1}{r}$ = $\dfrac{6}{24}$

$\dfrac{1}{r}$ = $\dfrac{1}{4}$

So,

r = 4.

Now,

$\small\triangle$ = √r×r1×r2×r3

=> 96.

(1).

r = △/s

=> 4 = 96/s

=> s = 24

(2).

r1 = △/(s-a)

=> 8 = 96/(24-a)

=> a = 12

(3).

r2 = △/(s-b)

=> 12 = 96/(24-b)

=> b = 16

(4).

r3 = △/(s-c)

=> 24 = 96/(24-c)

=> c = 20


Hence proved

Hope it helps u ♡♡


$\huge\mathfrak{Thank\: You}$