How to find the length X | Circle inside a right angle triangle | Geometry Problem

How to find the length X | Circle inside a right angle triangle | Geometry Problem
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[@dantallman5345]

Area of triangle, ht =15 and base =x, with an inscribed circle with radius 3.
A=(1/2)rP. And r=3. A=(1/2)(15)(x)…..set A=A hence P=(5)x.
P=15+12+x+(x-3)=24+(2)x…, hence (5)x=24+(2)x……x=8.

[@marcelowanderleycorreia8876]

Perfect Sir!!

[@quigonkenny]

Let the points of tangency between the circle and triangle ∆ABC be as follows: M on AB, N on BC, T on CA. Let the center of the circle be O.
Draw OM, ON, and OT. All 3 equal r, being radii of circle O, and As AB, BC, and CA are tangent to circle O at those three points, ∠OMB = ∠BNO = CTO = 90°. Therefore ∠NOM must equal 90° as well, so BNOM is a square with side length r = 3.
Ad MB = 3 and AB = 15, AM = 15-3 = 12. As TA snd AM are tangents to circle O that intersect at A, TA = AM = 12. As BN = 3 and BC = x, NC = x-3. As NC and CT are tangents to circle O that intersect at C, CT = NC = x-3. CT = x-3 and TA = 12, so CA = x-3+12 = x+9.
Triangle ∆ABC:
AB² + BC³ = CA²
15² + x² = (x+9)²
225 + x² = x² + 18x + 81
18x = 144
x = 144/18 = 8

[@muhammadayyaz3112]

if we find diagonal in rectangle
and by adding it into radius and using pythagorean theorem
we get x=6+3√2

[@CanalMiTube]

Another more direct path would be to equalize areas. AB*BC/2 = AF*FC. 15x/2 = 12(x-3). x=8.
Greetings.

[@expertonnn]

X'2+15'2=(x+9)'2....x=8

[@alexniklas8777]

Аналогично решил

[@WahranRai]

@@alexniklas8777 No, he used the variable c, lengthening the calculation !

[@prbprb2]

Call angle ACB = 2 phi. Then tan (2 phi) = 15/X , tan(phi) = 3/(X-3) . Eliminate phi: X =8

[@PS-mh8ts]

Or you can use the incircle radius formula like so:
incircle radius r(of a right-triangle ABC with ∠B=90°)=(AB+BC-AC)/2 -- (i)
AB=15, BC=x, and AC=√(15²+x²)=√(225+x²), and r=3. Using these in (i), we get:
[15+x-√(225+x²)]/2=3
i.e., 15+x-√(225+x²)=6
=> √(225+x²)=15+x-6=9+x
Squaring both sides, 225+x²=81+18x+x²
i.e., 144=18x or x=8

[@nunoalexandre6408]

Love it!!!!!!!!!!

[@channelsixtyeight068_]

By the time we got to C = 5, I forgot what the original problem was. 🤣 Love your videos.

[@oxlq]

Best solution; r=(a+b-c)/2
Then we will get a equality and after solving this x=6

[@wes9627]

x=15*tan[2*atan(1/4)]=8

[@yakupbuyankara5903]

X=8

[@skwest]

Lying in be, doin' it in my head... I'll go with:
X = 8
?
I'll check back later after some sleep.
Cheers!

[@User-jr7vf]

Youre lying

[@adgf1x]

x=15^1/2

[@makermaker11]

Its 8