Answer:
[tex]XY=5\sqrt{7}\;\; \sf cm[/tex]
Step-by-step explanation:
If YP = 4PX then the ratio of XP : PY = 1 : 4.
Let XP = x and PY = 4x, therefore the length of XY = 5x.
Let M be the midpoint of chord XY.
Therefore, if XY = 5x then XM = MY = 2.5x.
So PM = 1.5x.
[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]
Use Pythagoras Theorem to create two expressions for OM².
For right triangle OMP:
- a = MP = 1.5x
- b = OM
- c = OP = 6
[tex]\implies MP^2+OM^2=OP^2[/tex]
[tex]\implies (1.5x)^2+OM^2=6^2[/tex]
[tex]\implies 2.25x^2+OM^2=36[/tex]
[tex]\implies OM^2=36-2.25x^2[/tex]
For right triangle OMY:
- a = MY = 2.5x
- b = OM
- c = OY = 8
[tex]\implies MY^2+OM^2=OY^2[/tex]
[tex]\implies (2.5x)^2+OM^2=8^2[/tex]
[tex]\implies 6.25x^2+OM^2=64[/tex]
[tex]\implies OM^2=64-6.25x^2[/tex]
Use the method of substitution to solve for x:
[tex]\implies OM^2=OM^2[/tex]
[tex]\implies 36-2.25x^2=64-6.25x^2[/tex]
[tex]\implies 4x^2=28[/tex]
[tex]\implies x^2=7[/tex]
[tex]\implies x=\sqrt{7}[/tex]
As XY = 5x, then:
[tex]\implies XY=5\sqrt{7}\;\; \sf cm[/tex]
