Mathematics Question and Solution
Calculating r, radius of the quarter circle.
a = (r-2) units.
Therefore;
r² = b²+(r-2)²
r²-(r²-4r+4) = b²
b² = 4r-4
b = √(4r-4) units.
c = (r-4) units.
Therefore;
r² = d²+(r-4)²
r²-(r²-8r+16) = d²
d² = 8r-16
d = √(8r-16) units.
e² = 2r²
e = √(2)r units.
It implies;
e² = f²+2²
(√(2)r)²-2² = f²
f² = 2r²-4
f = √(2r²-4) units.
Notice;
b+d = f
√(4r-4)+√(8r-16) = √(2r²-4)
(4r-4)+2√((4r-4)(8r-16))+(8r-16) = 2r²-4
12r-16+2√((4r-4)(8r-16)) = 2r²
2√((4r-4)(8r-16)) = 2r²-12r+16
√((4r-4)(8r-16)) = r²-6r+8
(4r-4)(8r-16) = (r²-6r+8)²
32r²-64r-32r+64 = r⁴-6r³+8r²-6r³+36r²-48r+8r²-48r+64
32r²-96r+64 = r⁴-12r³+52r²-96r+64
32r² = r⁴-12r³+52r²
r⁴-12r³+52r²-32r² = 0
r⁴-12r³+20r² = 0
Dividing through by r², therefore;
r²-12r+20 = 0
Resolving the above quadratic equation via completing the square approach to get r, radius of the quarter circle.
(r-6)² = -20+(-6)²
(r-6)² = 16
r = 6±√(16)
r = 6±4
It implies;
r ≠ 6-4
r = 6+4
r = 10 units.
