The figure shows a square ABCD of side 6. It contains arcs BD and AC drawn with centers at A and D, respectively. If x is the radius of the circle tangent to AB and arcs AC and BD. What is the value of x?
+26367 The figure shows a square ABCD of side 6.
It contains arcs BD and AC drawn with centers at A and D, respectively.
If x is the radius of the circle tangent to AB and arcs AC and BD. What is the value of x?
\(\begin{array}{|lrcll|} \hline (1) & (6+x)^2 = (6-x)^2 + y^2 \\ (2) & (6-x)^2 = x^2 + y^2 \\ \hline (1)-(2): & (6+x)^2-(6-x)^2 &=& (6-x)^2-x^2 \\ & (6+x)^2 &=& (6-x)^2+(6-x)^2-x^2 \\ & (6+x)^2 &=& 2(6-x)^2-x^2 \\ & 36+12x+x^2 &=& 2(36-12x+x^2) -x^2 \\ & 36+12x+x^2 &=& 2*36-24x+2x^2 -x^2 \\ & 36+12x &=& 2*36-24x \\ & 36+36x &=& 2*36 \\ & 36x &=& 36 \\ & \mathbf{x} &=& \mathbf{1} \\ \hline \end{array}\)
