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# Geometry Problem

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In rectangle ABCD, AD = 1, P is on AB, and DB and DP trisect angle ADC. Write the perimeter of triangle BDP in simplest form as: w+(x*sqrt(y))/z, where w, x, y, z are positive integers. What is w + x + y + z?

Jan 31, 2018

#1
+8095
+4

DB and DP trisect angle ADC,   so   m∠ADP  =  m∠PDB  =  m∠BDC  =  90° / 3  =  30°

∠ABD  and  ∠BDC  are alternate interior angles, so they have the same measure.

If we draw a line from  P to  BD  that is perpendicular to side BD, we will have congruent triangles.

The angle measures of these triangles are  30°, 60°, and 90° .

The side opposite the  60°  angle  =  1 , so

the side opposite the 90° angle  =  2 / √3

perimeter of triangle BDP   =  2  +  2/√3  +  2/√3   =   2  +  4/√3   =   2  +  4√3 / 3

2 + 4 + 3 + 3   =   12

Jan 31, 2018

#1
+8095
+4

DB and DP trisect angle ADC,   so   m∠ADP  =  m∠PDB  =  m∠BDC  =  90° / 3  =  30°

∠ABD  and  ∠BDC  are alternate interior angles, so they have the same measure.

If we draw a line from  P to  BD  that is perpendicular to side BD, we will have congruent triangles.

The angle measures of these triangles are  30°, 60°, and 90° .

The side opposite the  60°  angle  =  1 , so

the side opposite the 90° angle  =  2 / √3

perimeter of triangle BDP   =  2  +  2/√3  +  2/√3   =   2  +  4/√3   =   2  +  4√3 / 3

2 + 4 + 3 + 3   =   12

hectictar Jan 31, 2018
#2
+101090
+3

Angle ADP  = 30°   and angle APD  = 60°

So.....since we have a 30-60-90 triangle......AP  =  1/√3   = √3/3

And PD  =   2 / √3  =  2√3 / 3

And angle DPB = 120°  and angle PDB = 30° ..... so   angle  PBD = 30°

So...triangle DPB  isosceles with  PD  = PB  =  2√3 / 3

And using the Law of Sines

DB/ sin(120)  = PB / sin (30)

DB  =  sin (120) * [ 2√3 / 3 ]  /  (1/2)  =    √3 * 2√3 / 3   =  2

So...the perimeter of triangle BDP  =

PD + PB + DB  =     4√3 / 3  +  2   =

2  +  4√3 / 3

So   w  = 2,  x = 4 , y  = 3  and z  = 3

And their sum is  12

Jan 31, 2018
#3
+101090
+1

I liked the way you did this, hectictar....the "congruent" triangles approach was a nice touch...!!!

Jan 31, 2018
#4
+8095
+3

Thanks!!

hectictar  Jan 31, 2018