find the radius of the circle inscribed in a triangle whose sides are 15cm, 17 cm and 8cm?
make a diagram
Triangle ABC with AB=17, BC=8 and AC = 15
Did you realize that your triangle is right-angled, with angle C = 90° ?
let the points of contact of the circle be
D on BC, E on AC, and F on AB
Two properties we can use ...
1. The centre of the incscribed circle lies on the angle bisectors of the triangle, and
2. BF=BC , AE=AF, and DC = EC, by the tangent properties
let the radius be r,
since the 90° is bisected, DC = r
then BD= 8-r, and of course by #2 BF=8-r
then AF= 9+r and AE = 15-r
I see two similar triangles at the top, so
r/(9+r) = r/(15-r)
solve for r
3cm
To find the radius of the circle inscribed in a triangle, you can use a formula called the inradius formula.
The inradius (r) of a triangle is given by the formula:
r = (√(s*(s-a)*(s-b)*(s-c))) / s
where "a", "b", and "c" are the lengths of the sides of the triangle, and "s" is the semiperimeter of the triangle, given by:
s = (a + b + c) / 2
Now let's use this formula to find the radius of the circle inscribed in the triangle with sides measuring 15 cm, 17 cm, and 8 cm:
a = 15 cm
b = 17 cm
c = 8 cm
s = (15 + 17 + 8) / 2 = 40 / 2 = 20 cm
Now, substituting these values into the inradius formula:
r = (√(20*(20-15)*(20-17)*(20-8))) / 20
= (√(20*5*3*12)) / 20
= (√(3600)) / 20
= 60/20
= 3 cm
Therefore, the radius of the circle inscribed in the given triangle is 3 cm.