A triangle has sides with lengths of 2, 9, and 8. What is the radius of the triangles inscribed circle?

Question

A triangle has sides with lengths of 2, 9, and 8.  What is the radius of the triangles inscribed circle?

Ella Armstrong · Accepted Answer

Radius of the inscribed circle #r=0.77(2dp)# unit.

sides of the triangle are #a=2 ; b=9; c=8#. Semi perimeter of the triangle #s=(2+9+8)/2=9.5 ; s-a= 7.5; s-b= 0.5 ; s-c= 1.5 :. A_t=sqrt(9.5*7.5*0.5*1.5)=7.31 sq.unit#. Where #A_t# is the area of triangle#=sqrt (s*(s-a)(s-b)(s-c))#. Radius of the inscribed circle #r=A_t/s= 7.31/9.5=0.77(2dp)#unit [Ans]

Allison Allen · Answer

undefined The radius $ r $ of the inscribed circle of a triangle can be calculated using the formula:

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

The semiperimeter $ s $ of the triangle is given by the formula:

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

where $ a $, $ b $, and $ c $ are the lengths of the sides of the triangle.

Given that the sides of the triangle are $ a = 2 $, $ b = 9 $, and $ c = 8 $, we can find $ s $ as:

$$ s = \frac{{2 + 9 + 8}}{2} = 9.5 $$

Next, we need to find the area $ A $ of the triangle using Heron's formula:

$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$

Substituting the values:

$$ A = \sqrt{9.5(9.5 - 2)(9.5 - 9)(9.5 - 8)} = \sqrt{9.5 \times 7.5 \times 0.5 \times 1.5} = \sqrt{53.4375} \approx 7.31 $$

Now, we can find the radius $ r $ of the inscribed circle using the formula mentioned earlier:

$$ r = \frac{A}{s} = \frac{7.31}{9.5} \approx 0.77 $$

Therefore, the radius of the triangle's inscribed circle is approximately $ 0.77 $ units.