Find the radius of the circle | Math Olympiad | Important Geometry Skills Explained

Find the radius of the circle | Math Olympiad | Important Geometry Skills Explained
Join this channel to get access to perks:
/ @mathbooster
MEMBERS OF THIS CHANNEL
••••••••••••••••••••••••••••••••••••••••
Mr. Gnome
Sambasivam Sathyamoorthy
ஆத்தங்கரை சண்முகம்
Εκπαιδευτήρια Καντά
AR Knowledge
堤修一
Sidnei Medeiros Vicente

[@اممدنحمظ]

تمرين جميل جيد . رسم واضح مرتب . شرح واضح مرتب . شكرا جزيلا لكم والله يحفظكم ويرعاكم ويحميكم جميعا. تحياتنا لكم من غزة فلسطين .

[@MathBooster]

Thank you 🙂

[@sarantis40kalaitzis48]

Right Triangles ABC and AMN are similar,since angle BAC is common. From ratios we have AP/AC=MP/BC so AP/AC=r/5 (1).Also right triangles CPN and CAD are similar,so CP/AC=PN/AD,so CP/AC=r/15 (2). Adding by parts (1) and(2) (AP/AC)+(CP/AC)=(r/5)+(r/15) so (AP+CP)/AC=(3r+r)/15 ,soAC/AC=4r/15 so 1=4r/15 so 4r=15 so r=15/4 (3).From triangles ABC and AMP similarity we have BC/MP=AC/AP ,so 5/r=2R/(R+10) with r=15/4 known from (3). So 5/(15/4)=2R/(R+10) so 4/3=2R/(R+10) so 4*(R+10)=2R*3 so 4R+40=6R so 2R=40 hence R=20 ._

[@alinayfeh4961]

similarth triangle ACD~ACB (1\3) AP/PM=AC/CB R+10/radius= 2R/5 in traingle CPN and CAD Are identical 15R-150=(2R)radius 2R/5=10+R/radius (R-10)=(r/3)(R+10) (R+10)=Rr+3⅓ R=20

[@Istaphobic]

Nice, I solved it.

[@klementhajrullaj1222]

How is the little r?!