#righttriangle

[num8ers]

[num8ers]

[logicxonomy]

Trisection of hypotenuse

In the given figure, the hypotenuse of a right-angled triangle is trisected by two circles. If the lengths of these segments are a, b, and c respectively. Then prove:- a/b×√[c/(a+b)]=2

Let the lengths of the sides PQ, QR, and PR of △PQR be y, x, and z respectively. The length of the radius of the smaller semicircle is r.

Let ∠QPR=θ

Therefore ∠QRP=90°-θ

Draw ST and QU

Then ∠STP and ∠QUR=90°

Theorem: The angle subtended by a diameter of a circle at any point on its boundary is 90°.

So, ∠PST=90°-θ and ∠UQR=θ

PQ is the tangent and PR is the secant of the larger circle, so by theorem

PQ²=PU×PR

y²=(a+b)×z

a+b=y²/z …….(i)

△QUR~△PQR (Similar Triangles)

UR/RQ=RQ/PR

c/x=x/z

c=x²/z …….(ii)

△PST~△PQU (Similar Triangles)

PT/TU=PS/SQ

a/b=2r/(y-2r) …….(iii)

Now draw VW, here V and W are the centers of the respective circles.

In the right triangle △VQW, apply Pythagoras’ Theorem:

VW²=VQ²+QW²

Put VW=r+x/2

VQ=y-r

QW=x/2

r=y²/(x+2y)

Put this value of r in equation (iii)

a/b=2r/(y-2r)

After solving, a/b=2y/x ……(iv)

Now put all the values in the given expression:

a/b×√[c/(a+b)]= (2y/x)×(x/y)=2

Hence Proved

— — — — — — — — — — — — — —

Click the given link for more concepts

In this chapter, we are going to discuss important right angle triangle theorems and properties along with their proofs.

Telegram Channel: Logicxonomy

[0acxin]

I drew a #brushstroke #design based on a specific shape(This time it is a #righttriangle ). The 2nd one is an explanation of it 以特定的圖形為主題，描繪了 #毛筆字 的 #設計 (這次是 #直角三角形 )。第二張是關於這個解釋 特定の図形をモチーフに、#筆文字 設計を描いてみました(今回は直角三角形)。2枚目はその解説です (Note(JA)-> https://note.com/nox_0/n/n76e4ed8bf43a )

By Nox-0

#書写 #アナログ画 #アート #calligraphy #brushlettering #art #ArtistOnInstagram #analogueart #art #artwork #ArtistOnTumblr #analogueart #書法 #藝術 #筆繪 #艺术 #书法 #peniculus ‎#קליגרפיה #schönschrift #芸術隠者 #ArtisticHermit #藝術隱士 #Nox0 #nox0art

https://www.instagram.com/p/Clihn8aS4g8/?igshid=NGJjMDIxMWI=

[diceojisan]

直角三角形 #righttriangle #art #artwork #drawing#artistoninstagram #artoftheday #illustration #illustragram #pen #penart #drawingartwork #ペン #ペン画 #アート https://www.instagram.com/p/CftqDFKBhm4/?igshid=NGJjMDIxMWI=

[yattooakcademi]

Design 3: Geometric Lenzes A design based in the world of geometry, mathematics, and 3D shapes. #Cone #RightTriangle & #Square #YATTOOAKCADEMYI https://www.instagram.com/p/BzjV4hYJfCV/?igshid=1bvh4hlfbdi6c

[mobilefidelity]

#sunrise through the #blinds #righttriangle #diagonallines (at San Marcos, California)

[neesh-art]

Must see in person— the most recent incarnation @alexpaik modular wall installation #righttriangle #curve catches the light beautifully in group exhibition “SkyLight” on view today @aquarius_studios_nyc