RO divides the rectangle into two congruent right triangles.
The area of the one triangle is equal half area of the rectangle.
Calculate the area of rectangle:
[tex]A_R=lw\\l=12,\ w=5\\\\A_R=12\cdot5=60[/tex]
The area of right triangle:
[tex]A_T=\dfrac{1}{2}A_R\to A_T=\dfrac{1}{2}\cdot60=30[/tex]
Use the Pythagorean theorem to calculate the length of RO:
[tex]|RO|^2=5^2+12^2\\\\|RO|^2=25+144\\\\|RO|^2=169\to|RO|=\sqrt{169}\to|RO|=13[/tex]
The formula of an area of this right triangle is:
[tex]A_T=\dfrac{1}{2}|RO||PA|[/tex]
Therefore we have the equation:
[tex]\dfrac{1}{2}(13)|PA|=30\qquad|\cdot2\\\\13|PA|=60\quad|:13\\\\|PA|=\dfrac{60}{13}\\\\|PA|\approx4.62[/tex]
