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# triangle

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In right triangle ABC, leg AC is 2 cm and leg BC is 6 cm. A square is constructed of hypotenuse AB, the hypotenuse being one side of the square. What is the distance, in cm, from point C to the intersection point of the two diagonals of the square?

A) 4√3

B) 4√2

C) 6√2

D) 5√3

E) 5√2

jonathanxu999  Dec 2, 2017
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#1
+82676
+2

The answer is "B"....I'll fill in the blanks, later.....

CPhill  Dec 2, 2017
edited by CPhill  Dec 4, 2017
#2
+82676
+3

Put A at (0,2), C at (0,0), and B at (6,0)

AB  is   √40  units in length

Construct two lines perpendicular to AB  through A  and B

The equation of the line through A  is  y = 3x + 2 .......the of the line through B is

y = 3x  - 18

Constuct two circles with radiuses of √40  centered at A  and B

The equation of the circle centered at A  is  x^2 + (y - 2)^2  = 40

The equation of the circle centered at B  is  (x - 6)^2 + y^2  = 40

And we can find the intersection of the circle centered at A and the perpendicular line to AB passing through A  thusly :

x^2 + (3x + 2 - 2)^2   =  40

x^2 + 9x^2  =  40

10x^2  =  40

x =  2   and y  = 3(2) + 2  = 8

So  the intersection is at  (2, 8)....label this point D

Likewise....we can find the intersection of the cirlce centered at B and the perpendicular line to AB passing through B  thusly :

(x - 6)^2  + (3x - 18)^2  = 40

(x - 6)^2 + 9 ( x  - 6)^2  = 40

10 (x - 6)^2  = 40

(x - 6)^2  =  4

x - 6  = 2

x = 8    and y  = 3(8) - 18   =  6

So the intersection  is  (8, 6).....label this point E

And  DB  will be one diagonal of the square and AE will be the other diagonal

And the equation of DB  is   y  = -2x + 12

And the eqation of AE  is y =  (1/2)x  + 2

And we can find the inersection of the diagonals, thusly

-2x + 12  = (1/2)x + 2

10 = 2.5x

4  = x      and  y  =  -2(4) + 12   = 4

So the intersection of the diagonals is at (4,4)

And the distance between this point and C  =  √ [ 4^2 + 4^2]   = √32  = 4√2

CPhill  Dec 3, 2017
edited by CPhill  Dec 4, 2017
#3
+11261
+1

I wanted to see how Chris got the EXACT answer with a radical ....I got an equivalent numerical answer using the Law of Cosines (twice)   See handwritten diagram:

ElectricPavlov  Dec 3, 2017
edited by ElectricPavlov  Dec 3, 2017

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