Mathematics Question and Solution
Calculating r, radius of the circle.
Let the small inscribed right-angled triangle shortest length be x.
3 ~ 4
x ~ a
Cross Multiply.
3a = 4x
a = ⅓(4x) units.
a is the big inscribed right-angled triangle shortest length.
b²+x² = 3²
b = √(9-x²) units.
3 ~ 4
√(9-x²) ~ c
Cross Multiply.
3c = 4√(9-x²)
c = ⅓(4√(9-x²)) units.
It implies;
3x = x+c
3x = x+⅓(4√(9-x²))
9x = 3x+4√(9-x²)
6x = 4√(9-x²)
3x = 2√(9-x²)
9x² = 4(9-x²)
9x² = 36-4x²
13x² = 36
x² = 36/13
x = √(36/13)
x = 6√(13)/13 units.
x = 1.66410058868 units.
Recall.
a = ⅓(4x)
And x = 1.66410058868 units.
a = ⅓*4*1.66410058868
a = 2.2188007849 units.
b = √(9-x²)
And x = 1.66410058868 units.
b = √(9-1.66410058868²)
b = 2.49615088301 units.
c = ⅓(4√(9-x²))
And x = 1.66410058868 units.
c = ⅓*4√(9-1.66410058868²)
c = 3.32820117735 units.
d = ½(x+c)
d = ½(1.66410058868+3.32820117735)
d = 2.49615088302 units.
e = ½(a+b)
e = ½(2.2188007849+2.49615088301)
e = 2.35747583396 units.
f = e-a
f = 2.35747583396-2.2188007849
f = 0.13867504906 units.
Therefore r, radius of the circle is;
r² = f²+d²
r² = 0.13867504906²+2.49615088302²
r² = 6.25000000003
r = √(6.25000000003)
r = 2.5 units.
Or
3²+4² = 4r²
25 = 4r²
r² = 25/4
r = √(25/4)
r = ½(5)
r = 2.5 units.
