SOLUTION: The dimensions of rectangle ABCD are AB=12 and BC=16. Point P is marked on side BC so that BP=5, and the intersection of AP and BD is called T. Find the lengths of the four segme

Algebra ->  Formulas -> SOLUTION: The dimensions of rectangle ABCD are AB=12 and BC=16. Point P is marked on side BC so that BP=5, and the intersection of AP and BD is called T. Find the lengths of the four segme      Log On


   

Question 149701: The dimensions of rectangle ABCD are AB=12 and BC=16. Point P is marked on side BC so that BP=5, and the intersection of AP and BD is called T. Find the lengths of the four segments TA, TP, TB, and TD.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
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%2F16=x%2F%2813-x%29 and 5%2F16=y%2F%2820-y%29


So let's solve for x:

%285%29%2F%2816%29=%28x%29%2F%2813-x%29 Start with the first ratio


%285%29%2813-x%29=%28x%29%2816%29 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=%28-65%29%2F%28-21%29 Divide both sides by -21 to isolate x.


x=65%2F21 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:

%285%29%2F%2816%29=%28y%29%2F%2820-y%29 Start with the second ratio


%285%29%2820-y%29=%28y%29%2816%29 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=%28-100%29%2F%28-21%29 Divide both sides by -21 to isolate y.


y=100%2F21 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.