Two Triangle Problems
Date: Tue, 11 Jun 1996 21:28:40 -0400 (EDT) From: Anonymous Subject: two triangle problems Here are two triangle problems I can't solve. 1. In triangle ABC, angle A is trisected by segments AE and AD where points E and D lie on BC. BD = 2, DE = 3, and EC=6. Find the length of the shortest side of triangle ABC. 2. If triangle ABC is reflected in its median AM, and AE = 6, EC = 12, BD = 10, and AB = k(sqrt3), compute k. A | \ /|\ \ B' / | \ E B---D---M-----C | / C' Sorry about the diagram but I hope you can see ABC and AB'C', D, and E are intersection points of AC' and BM , and AC and B'M respectively. Thanks for your help, Michael
Date: Thu, 13 Jun 1996 17:26:57 -0400 (EDT) From: Dr. Anthony Subject: Re: two triangle problems I will look at question (2) tomorrow, but here is question (1) to be going on with. (1) It would be wise to draw an approximate diagram for question(1) and refer to it as I go through the maths. It is unfortunately a rather long-winded explanation. In triangle ABD sin(A/3)/2 = sin(B)/AD In triangle ABE sin(2A/3)/5 = sin(B)/AE Equate the two values of sin(B) to get AD*sin(A/3)/2 = AE*sin(2A/3)/5 = 2AE*sin(A/3)cos(A/3)/5 cancel the sin(A/3) on both sides to get cos(A/3) = (5/4)(AD/AE) Now using the angle-bisector theorem on triangle ADC, we have AC/AD = 6/3 = 2 and using same theorem on triangle ABE we have AE/AB = 3/2 Multiply these to get (AC/AB)(AE/AD) = 3 or AD/AE = (1/3)(AC/AB)...(1) Now we need a further relationship, and we get this from further applications of the sine rule. sin(B)/AC = sin(C)/AB so sin(B)/sin(C) = AC/AB ....(2) also sin(B)/AE = sin(2A/3)/5 and sin(C)/AD = sin(2A/3)/9 Divide these and cancel sin(2A/3) to get sin(B)/sin(C) = (9/5)(AE/AD) = AC/AB from (2) above. Substitute for AC/AB in (1) above We have AD/AE = (1/3)(9/5)(AE/AD) (AD/AE)^2 = (3/5) AD/AE = sqrt(3/5) Now cos(A/3) = (5/4)sqrt(3/5) = (1/4)sqrt(15) At this stage I shall be working with the value of angle A, B, C in degrees. We have sin(A/3) = sqrt(1 - 15/16) = sqrt(1/16) = 1/4 So A/3 = 14.4775 degrees and A = 43.4325 degrees. We had sin(B)/sin(C) = (9/5)AE/AD = (9/5)sqrt(5/3) = sqrt(27/5) . sin(B)/sin(C) = 2.32379 Also B + C = 180 - 43.4325 = 136.5675 degrees B = 136.5675-C Sin(136.5675-C) = 2.32379sinC sin(136.5675)cos(C) - cos(136.5675)sin(C) = 2.32379sin(C) 0.6875cos(C) + 0.7262sin(C) = 2.32379sin(C) 0.6875cos(C) = 1.597605sin(C) and so tan(C) = 0.43033 and angle C = 23.28374 degrees Finally AB/sin(C) = 11/sin(A) AB = (11*sin(C))/sin(A) = (11*sin(23.28374))/sin(43.4325) AB = 6.32456 This is opposite the smallest angle so is shortest side -Doctor Anthony, The Math Forum
Date: Thu, 13 Jun 1996 21:00:02 -0400 (EDT) From: Dr. Anthony Subject: Re: two triangle problems I sent the answer to question (1) yesterday, and here, as promised, is question (2). Again draw the diagram and refer to it as I work through the question. Using the angle-bisector theorem with triangle ADC, we see that DM:MC is in the ratio 18:6 = 3:1 But M is mid-point of BC, so BD+DM = MC 10+DM = 3DM 10 = 2DM and so DM = 5, MC = 15 We can now find angle C from the triangle EMC since all three sides are known.(5,12,15 with 5 opposite angle C) Using the cosine formula we get: cos(C) = (15^2+12^2-5^2)/(2*15*12) = 344/360 = 43/45 Now apply the cosine rule to triangle ABC.(BC=30, AC=18) AB^2 = 30^2 + 18^2 - 2*30*18*cos(C) = 900 + 324 - 2*30*18*43/45 = 900 + 324 - 1032 = 192 = 64*3 And taking square roots AB = 8*sqrt(3) So, in the question k = 8 -Doctor Anthony, The Math Forum
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