Question

Find the length of the line PA.

Answer 1

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:

`A_R=lw\\l=12,\ w=5\\\\A_R=12\cdot5=60`

The area of right triangle:

`A_T=(1)/(2)A_R\to A_T=(1)/(2)\cdot60=30`

Use the Pythagorean theorem to calculate the length of RO:

`|RO|^2=5^2+12^2\\\\|RO|^2=25+144\\\\|RO|^2=169\to|RO|=√(169)\to|RO|=13`

The formula of an area of this right triangle is:

`A_T=(1)/(2)|RO||PA|`

Therefore we have the equation:

`(1)/(2)(13)|PA|=30\qquad|\cdot2\\\\13|PA|=60\quad|:13\\\\|PA|=(60)/(13)\\\\|PA|\approx4.62`