Question 149701
First draw the rectangle with the point P

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Now draw in the segments AP, BD, and PD. Take note of the unknown variables I'm assigning. If you look closely, you'll notice that a trapezoid forms due to these extra lines segments. Through the use of pythagoreans theorem, we get the length of AP of 13 and BD of 20



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Now, it turns out that the ratio of the parallel sides 5 and 16 are the same as the ratio of the lengths of the cut diagonals. So the following ratios are true:


{{{5/16=x/(13-x)}}} and {{{5/16=y/(20-y)}}}



So let's solve for x:


{{{(5)/(16)=(x)/(13-x)}}} Start with the first ratio



{{{(5)(13-x)=(x)(16)}}} Cross multiply



{{{65-5x=16x}}} Distribute and multiply.



{{{-5x=16x-65}}} Subtract {{{65}}} from both sides.



{{{-5x-16x=-65}}} Subtract {{{16x}}} from both sides.



{{{-21x=-65}}} Combine like terms on the left side.



{{{x=(-65)/(-21)}}} Divide both sides by {{{-21}}} to isolate {{{x}}}.



{{{x=65/21}}} Reduce.



So the approximate length of x is 3.095 units. This means that TP is 3.095 units. This means that the other length is {{{13-3.095=9.905}}}. So TA is 9.905 units.



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Now let's solve for y:


{{{(5)/(16)=(y)/(20-y)}}} Start with the second ratio



{{{(5)(20-y)=(y)(16)}}} Cross multiply



{{{100-5y=16y}}} Distribute and multiply.



{{{-5y=16y-100}}} Subtract {{{100}}} from both sides.



{{{-5y-16y=-100}}} Subtract {{{16y}}} from both sides.



{{{-21y=-100}}} Combine like terms on the left side.



{{{y=(-100)/(-21)}}} Divide both sides by {{{-21}}} to isolate {{{y}}}.



{{{y=100/21}}} Reduce.



So the approximate length of y is 4.762 which means that the other length is {{{20-4.762=15.238}}}. So TB is 4.762 units and TD is 15.238 units.