A Very Nice Geometry Problem From China | 2 Different Methods

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/ @mathbooster

[@jimlocke9320]

Using the designations give at 0:45, construct OC and OP. Note that ΔOPC and ΔOSC are congruent by side-side-side. Therefore

[@marioalb9726]

tanα = 2/5 --> α=21,8014°
b = 5+2 = 7 cm
x = c - 5 = b/cos2α - 5 = 14/3 cm
A = A₁-A₂ = 5.x - πR² = 70/3-4π
A = 10,767 cm² (Solved √)

[@marioalb9726]

tanα = 2/5 --> α=21,8014°
b = 5+2 = 7 cm
h = b tan2α = 20/3 cm
A = ½bh - πR² = ½ 7 20/3 - π2² = 70/3-4π
A = 10,767 cm² (Solved √)

[@WahranRai]

Math Booster you must be rigorous
You call S a vertex of the square and then S as semi-perimeter
In EUROPE and in the rest of the world, the radius is R or r (gamma design angle in triangle in general)

[@Grizzly01-vr4pn]

Kind of valid first point, although it's impossible to tell if he's using 'S' for one and 's' for the other.
For your second point, he is actually writing 'r' believe it or not, it's just that his appalling handwriting makes it look like a lower-case gamma. Very annoying to see.

[@michaeldoerr5810]

The shaded area is 70/3-4pi units square. I have noticed that the second method makes use of the same circle method as before and uses the semiperimeter. How many problems that you know require BOTH that circle theorem and the semiperimeter???

[@devondevon4366]

70/3 - 4pi or 10.767
7^2 + ( n + 2)^2 = (n+5)^2
49 + n^2 + 4 + 4n =n^2 + 10n + 25
28 = 6n
14/3 = n
bases of triangle are (14/3 + 2 ) and 7
or 20/3 and 7
Area of triangle is 10/3 *7 = 70/3 or 23.333
Area of circle = 2^2 pi or 4pi
Hence, shaded area = 70/3 - 4pi or 10.767

[@prossvay8744]

Shaded area =1/2(20/3)(7)-4π=70/3-4π.❤

[@himo3485]

(2+5)²+(2+x)²=(x+5)² 49+4+4x+x²=x²+10x+25 6x=28 x=14/3
2+14/3=6/3+14/3=20/3
Shaded area = 20/3*7*1/2 - 2*2*π = 70/3 - 4π

[@giuseppemalaguti435]

(2+x)^2+7^2=(5+x)^2..4x+53=10x+25..x=28/6=14/3...Ared=(2+14/3)7/2-4π=20*7/6-4π=70/3-4π

[@sorourhashemi3249]

Thanks. Easy

[@RealQinnMalloryu4]

(2)^2=4A/ (5)^2=25O/ {4A/+25O/}=29AO/ 180°ABC/29AO=6.6ABCAO/3^2.3^2 1^1.3^2 (AO/ABC ➖ 3AO/ABC+2).

[@imetroangola4943]

This math, my friend, doesn't exist! What planet are you from? Are you a UFO by any chance?