find the radius of a circle that is the tangent to all the three side of the triangle ABC given that the triangle hve lenght 7,24 and 25
help?
http://mathcentral.uregina.ca/qq/database/qq.09.06/s/maria2.html
make a sketch
you will need the incentre of the triangle.
This consists of the intersection of the angle bisectors
label the triangle ABC, with the incentre O.
Sketch in the three radii, hitting sides AB, BC, and CA at P, Q, R respectively.
You should see that we have 3 pairs of congruent triangles, and
AP = AR = x
BP = BQ = y
CQ = CR = z
so x+y = 24 *
x+z = 25 **
y+z = 7 ***
*** - ** ----> z - y = 1
z = y+1
sub into ***
y + y+1 = 7
2y = 6
y = 3
z = 3+1 = 4
x+3 = 24
x = 21
We can find angle A by the cosine law:
7^2 = 24^2 + 25^2 - 2(24)(25)cosA
cosA = (576+625-49)/(1200)
A = appr 16.26°
Now in triangle AOP, angle P = 90°, angle OAP = 8.13°, AP = 21, OP = r
r/21 = tan 8.13
r = 21tan8.13° = 3
the radius is 3 units
To find the radius of the circle that is tangent to all three sides of the triangle ABC, we can use the formula:
r = (a + b - c) / 2
where r is the radius of the circle, and a, b, and c are the lengths of the triangle's sides.
In this case, the lengths of the sides of triangle ABC are 7, 24, and 25.
Let's substitute these values into the formula:
r = (7 + 24 - 25) / 2
r = 6 / 2
r = 3
Therefore, the radius of the circle is 3.
To find the radius of the circle that is tangent to all three sides of triangle ABC, we can use the concept of the inradius of a triangle. The inradius is the radius of a circle that is inscribed within a triangle, which is tangent to all three sides.
We can use the formula for the inradius of a triangle:
inradius = (area of the triangle) / (semiperimeter of the triangle)
First, let's find the semiperimeter of the triangle ABC. The semiperimeter is half the sum of the lengths of all three sides:
semiperimeter = (7 + 24 + 25) / 2 = 56 / 2 = 28
Next, let's find the area of the triangle using Heron's formula:
area = √(s(s-a)(s-b)(s-c))
where s is the semiperimeter, and a, b, and c are the lengths of the sides.
area = √(28(28-7)(28-24)(28-25))
= √(28 * 21 * 4 * 3)
= √(7056)
= 84
Now, we can find the inradius:
inradius = area / semiperimeter
= 84 / 28
= 3
Therefore, the radius of the circle that is tangent to all three sides of triangle ABC is 3.