Geometry Problem

Click the figure below to see the complete problem 505 about Right Triangle, Cevian, Sum of Segments, Angles.

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Complete Problem 505

Level: High School, SAT Prep, College geometry

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## Wednesday, August 18, 2010

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Problem 505: Right Triangle, Cevian, Sum of Segments, Angles

See also:

Complete Problem 505

Level: High School, SAT Prep, College geometry

## Link List

Online Geometry theorems, problems, solutions, and related topics.

Geometry Problem

Click the figure below to see the complete problem 505 about Right Triangle, Cevian, Sum of Segments, Angles.

See also:

Complete Problem 505

Level: High School, SAT Prep, College geometry

Labels:
angle,
cevian,
right triangle,
sum of segments

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With the given data is it possible to prove that BD bisects /_B ? Easy to do this by trigonometry but how do we do it by plane geometry alone?

ReplyDeleteOf course, this makes x=15 deg.

Vihaan

draw DBM = x => BM = a, get AH = a + d , H on AC

ReplyDelete=> CH = DM => HM = d => MA = a => MBA = 2x, B = 6x

=> x = 15°

To c.t.e.o

ReplyDeleteIt not clear to me why CH=DM in your solution. Please explain.

Peter Tran

geting AH = a + d give us ang HBC = x

ReplyDeletedrawing DBM = x => ▲DMB = ▲HBC

(BM = BC = a, MCB = CMB, MBD = CBH = x : ASA )

In triangle MBC altitude is bisector, giving BM = a

To: c.t.o.e.

ReplyDeleteWhat a nice soluion, well done!

Ajit

To Joe

ReplyDeleteThanks

If BG is altitude of ABC than BG is median of MBC,DBH

=> MD = HC as diferences of equal parts

( I liked your solution of P 167 )

To c.t.e.o

ReplyDeleteYour solution is the most beautiful solution I've ever seen.

Trisect < CBD by BE and BF, E and F on AC and BE thus being an altitude as well

ReplyDeleteLet CE = e so that DE = EF = d-e and CF = 2e-d

< AFB = 90-x = < ABF so AB = AF = a+d = b - (2e-d) hence e = (b-a)/2......(1)

Now CB is tangential to Tr. ABE at B so,

a^2 = eb

Substituting for e in (1),

b(b-a)/2 = a^2 hence b^2 - ab - 2b^2 = 0

Solving the quadratic and rejecting the negative solution we have b = 2a hence 2x = 30 and x = 15

Sumith Peiris

Moratuwa

Sri Lanka