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What's the value of AB? - July 1-7, 1996 Next Problem || Previous Problem || Contents || All Problems || Search POWs In parallelogram ABCD, the bisector of angle <ABC intersects AD at P. If PD=5, BP=6, and CP=6, what is the value of AB? School's out for the summer in the continental United States, so we have the pleasure of seeing answers arriving from the Far East - notably Queen Elizabeth School (Hong Kong), Raffles Girls' School (Singapore), and Iolani School (Hawaii). Here's a representative sample of correct solutions: Chui Yuk Man Grade: Secondary 6 School: Queen Elizabeth School(Hong Kong) Since BP bisects <ABC, <ABP = <PBC Since BC//AP ,<APB = <CBP Since BP = CP = 6 ,<PBC = <BCP Therefore triangle ABP is similar to triangle PBC (equiangular) So, AP/BP = PC/BC AP/6 = 6/BC AP/6 = 6/(5+AP) 5AP+AP^2 = 36 (AP+9)(AP-4) = 0 AP=4cm (since -9 is rejected) Since AP = AB , AB = 4.00cm Hang Xue Min Grade: 8 School: Raffles Girls' School (Secondary) Answer: The value of AB = 4 Since angle ABP = angle CBP, then angle PCB = angle CBP (isosceles triangle) and angle APB = angle CBP (alternate angles). Triangle APB will also be an isosceles triangle, similar to triangle CBP because the base angles have the same value. Now, let the value of AD be x. Therefore, AP=x-5 and AB will also be x-5. AB PB ---- = ---- PB CB x-5 6 ---- = ---- 6 x 2 x -5x -36 = 0 x = 9 or -4. x = -4 cannot be the solution because it is negative. Therefore, the solution is x = 9 and the value of AB = 9-5 = 4. Ling Qi Grade: 11 School: LSMSA Answer: All right, 4 sider ABCD... Let's set half of angle ABC = x, thus angle ABP = x, CBP = x BPC = 180-2x BCP = x BAD = 180-2x because it is a parallelogram. We conclude triangle ABP and BCP are similar. From that 2 angles from each trangle are equal to the relative ones on the other, and from here we can set AB = y, and we know that AB:BP = BP:BC, and BP = 6 and BC = AP+DP, but since AP = AB and PD = 5 thus BC = AB+5 Thus... y:6 = 6:y+5 y^2+5y-36 = 0, y = 4 or y = -9, and y = 4, Formal answer follows... *Drum roll please* The Final Answer is... 4! Done. Yang Yajun Grade: 7 School: Raffles Girl's Secondary Answer: Angle APB = Angle PBC (alternate angles) Angle ABP = Angle APB Therefore triangle ABP is an isoceles triangle. If I reproduce triangle ABP to form a parallelogram, I can get the dimensions of triangle ABP as 6 cm as base and 3 cm as height because I draw another 6 cm diagonal. Therefore, the height of triangle ABP is 3 cm. The area of triangle ABP = 1/2 * 6 * 3 = 9 sq cm Therefore the area of the parallelogram is 18 sq cm. The angles of the parallelogram are nearly like a square. The closest square to 18 is 4 * 4 = 16. If I establish the base of the parallelogram (AP) as 4 cm, the height is 4.5 cm. Since ABP is an isosceles triangle, therefore AB is 4 cm. Jason Yeung Grade: 9 School: Iolani School Answer: AB is 4 When you bisect angle ABC, angle ABP and angle PBC are equal. If BP=6 and CP=6, then triangle BPC is an isosceles triangle. Therefore, angle PBC and angle PCB are congruent (Isosceles Triangle Theorem). Angle ABP will also be congruent (Transitive Prop.). Since this is a parallelogram, segments AD and BC are parallel. With segment BP (transversal) intersecting them, angle APB and angle PBC are congruent because they are alternate interior angles. This proves that measurements of angles PBC = PCB = ABP = APB. I have sufficient information to prove that triangles ABP and PBC are similar through the Angle Angle Similarity Theorem. If I assume AB and AP are x, then AD will be x+5, and so will BC because they are parallel. I can then set up the ratio, x 6 - = --- 6 x+5 2 2 x +5x = 36, x +5x-36 = 0, (x+9)(x-4) = 0 x cannot be a negative number, so x is 4. And since AB = x, so AB = 4 Ken Duisenberg School: Stanford University, '91 Answer: AB = 4 Solution: Let angle ABC = 2x. Then ABP = PBC = x and PAB = 180-2x. In triangle PAB, Angle APB = 180 - PAB - ABP = x. So triangle PAB is isosceles, with base BP=6. Since BP=PC=6, triangle BPC is also isosceles with base BC, and sides of length 6. So angle BCP = x, and angle BPC = 180-2x. By similar angles, triangles ABP and BPC are similar. The following ratio leads to the solution. (Note that AB=AP, and AD=BC): AB 6 ----- = ----- --> (AB)^2 + 5(AB) - 36 = 0 6 AB+5 Using the quadratic formula for AB, we find AB = 4.

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