A triangle has corners at $(2, 1)$ $(2 , 1 )$ , $(3, 4)$ $(3 ,4 )$ , and $(1, 2)$ $(1 ,2 )$ . What is the radius of the triangle's inscribed circle?

Question

1438 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-12T19:26:43

$0.5402 (4 d p)$ $0.5402 (4dp)$ .

Let us name the given points $A (2, 1), B (3, 4) and C (1, 2)$ $A(2,1), B(3,4) and C(1,2)$ .

Using Distance Formula, we find that,

$A B^{2} = {(2 - 3)}^{2} + {(1 - 4)}^{2} = 1 + 9 = 10$ $AB^2=(2-3)^2+(1-4)^2=1+9=10$

Similarly, $B C^{2} = 8, and, A C^{2} = 2$ $BC^2=8, and, AC^2=2$ .

We notice that, $B C^{2} + A C^{2} = A B^{2},$ $BC^2+AC^2=AB^2,$ which means that

$Delta ABC$ is right-angled having $AB$ as its Hypotenuse.

We know from Geometry that, the Inradius $r$ of a right $Delta$ is

$1/2$ (sum of the lengths of sides making the right angle-length of

hypo.)

$:. r=1/2[sqrt2+sqrt8-sqrt10]=1/2(3sqrt2-sqrt10)$

$=sqrt2/2(3-sqrt5)~~1.4142/2(3-2.2361)~~0.5402(4 dp)$ .

A triangle has corners at (2 , 1 )(2,1)(2 , 1 ), (3 ,4 )(3,4)(3 ,4 ), and (1 ,2 )(1,2)(1 ,2 ). What is the radius of the triangle's inscribed circle?