Question

in triangle ABC r=1 R=4 ∆=8 then value of ab+bc+ca​

Answer 1

Answer:

x is equal to 81

Step-by-step explanation:

hence by using properties of triangles formulaes we can solve this

Answer 2

The value of ab+ bc + ca is 81

Step-by-step explanation:

In a triangle ABC,

Inradius formula is,

$r=\frac{\text{Area of the triangle}}{\text{Semi perimeter}}$

While,

Circumcentre formula,

$R=\frac{AB\times BC\times CA}{4\times \text{ Area of the triangle}}$

If Area,  ∆=8 square unit, r = 1 unit, R = 4 unit,

a = BC, b = AC and c = AB,

Semi perimeter of the circle = s

From first formula,

$\implies 1 =\frac{8}{s}$

$\implies s=8$

$\implies \frac{a+b+c}{2}=8$

$\implies a+b+c=16$              ............(1),

From second formula,

$4 =\frac{abc}{32}$

$\implies abc = 128$             ............(2),

Now, by the heron's formula,

Δ² = s(s-a)(s-b)(s-c)

64 = 8(8-a)(8-b)(8-c)

8 = (8-a)(8-b)(8-c)

$8=512-64b-64a+8ab-64c+8bc+8ac-abc$

$8 = 512 - 64(a+b+c) + 8(ab+bc+ca)-abc$

By substituting values,

$8=512 - 64(16) + 8(ab+bc+ca)-128$

$8= 512 - 1024 + 8(ab+bc+ca)-128$

$8=-640 + 8(ab + bc+ca)$

$8+640 = 8(ab+bc+ca)$

$648=8(ab+bc+ca)$

$\implies ab+bc+ca=81$

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