civonamzuk

Triangle ABC has circumcenter O. If AB = 10 and [OAB] = 50 find the circumradius of triangle ABC.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let ΔABC be an isosceles triangle with base AB = 10.

Let the midpoint of AB be M

 OM = (2 * 50) / 10 = 10

Circumradius    R = sqrt(OM² + AM²)

R = √125

Big circle radius => R

Small circle radius => r

Distance between the centers => m

? = n

n = R - r - m 

A right-angled triangle has a perimeter of 1276 and the longest side of 533.  What is the length of the inradius?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

P = 1276          s = 1276 / 2           Hypotenuse c = 533

Short leg = a         long leg = b            a + b = 743

a² + (743 - a)² = c²

a = 308           b = 435

A = 1/2(ab)

Inradius     r = A / s = 105

Perimeter = 1/6(10pi) + 10

This is very simple...

Let triangle ABC be the right triangle with angle A = 90 degrees.

[AXY] = 1/2(AX * AY) = 16

[ABC] = 1/2(AB * AC) = 35

[AXY] / [ABC] = 16 / 35 

sin(∠CAE) = 0.936  = 117/125 

Neat job, MPS!!! 

~~~~~~~~~~~~~~~~~~~~~

5a/3 + a + 3a/5 = √34

sin(∠BAC) = 3/5

sin(∠BAD) = 1

sin(∠CAE) = 0.936  ≈ 15/16

If a, b, and c are positive integers satisfying ab + c = bc + a = 13  and ac + b = 41, what is the value of a + b + c?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

b = 1

ac + 1 = 41

ac = 40

a + c = 13

(40/c) + c = 13

c = 8,   c = 5

a + b + c = 14

x = 32 / 4