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# Geometry Problem

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A semicircle inscribed inside a quarter circle. Find the length of a chord of the quarter circle that is also a tangent of the semicircle.

Jan 9, 2024

#1
+37045
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The answer I get is 36 cm ...... It took a while for me to figure a way to get the answer.....there is likely an easier way but here is my (long -hopefully-correct) solution:

See image :

Use arc tan to find the 63.435 angles then find the 53.13 angle     ( 63.435+63.435 + 53.13 = 180 ° )

then use sin and cos functions for angle 53.13 °  to find the coordinates of T

then use the coordinates of B and T to find the equation of the line BP to be   y = - 3/4 x + 30

the equation of the Quarter circle is    x^2 + y^2 = 900    or    y = sqrt(900-x^2)

equate this to the line equation and solve for the 'x' coordinate of  point P     then use the line equation and this value of the 'x' coordinate  to find the value of the 'y' coordinate for point P

Now you have the coordiantes of the two endpoints of the chord line

B    and   P  ====>    then finish by using the distance formula to find  BP = 36 cm

There really must be an easier way !!!    Whew !

Jan 9, 2024
#2
+129829
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I don't find an easier way, either, EP.

I did it with just some  Geometry

Here's my solution  (but it  might not  be  any  easier)

Like EP, I found BT to be 30

We can find the coordinates of T  by finding the intersection of two circles  (this is the key !!!)

One with a center of (0,30) and a  radius of 30

One with a center of (15,0) and a  radius of 15

The equations are

x^2  + ( y - 30)^2  = 900    →  x^2 + y^2 - 60y   =  0   (1)

(x - 15)^2 + y^2  = 225     →  x^2 - 30x + y^2  = 0     (2)

Subtract these

-60y + 30x  =  0

x = 2y

Sub  this into (2)

4y^2 - 60y + y^2  = 0

5y^2 -60y  =  0

y - 12  = 0

y =12

x = 2(12)   = 24

T = (24,12)

Slope   of   line through BP = ( 30 - 12) / ( 0 - 24) = -3/4

Equation of this  line  is  y =(-3/4)x + 30

Now, as EP  said, put this into  the  equation of the circle  x^2 + y^2  =900  and find the x coordinateof P  as  28.8

And  -(3/4)(28.8) + 30  =   8.4 = the y coordinate of P

TP = sqrt [ (28.8 -24)^2  + (12 - 8.4)^2 ]  = sqrt [ 4.8^2  + 3.6^2 ]  =  sqrt (36)   = 6

Nice job, EP!!!

Jan 9, 2024
#3
+37045
+1

Thanx, Chris !

I was wondering if I had this correct.....now I know it likely is correct.....or maybe we BOTH got it wrong ! HaHa

~EP

ElectricPavlov  Jan 10, 2024