Given: △KLM, KM=48, LD=16 LD ⊥ KM , NOPS - rectangle NO:OP=5:9 Find: NO, OP

Question

Given: △KLM, KM=48, LD=16

       

LD

 ⊥

KM

  , NOPS - rectangle

       NO:OP=5:9

    Find: NO, OP

Answer 1

Triangle LOP is similar to triangle LKM because segment OP runs parallel to KM ( due to all the right angles) and so splits the original triangle, LKM, proportionately. So the 16/48 ( or 1:3 ) ratio in the original triangle holds true for any other pair of similar triangles in the diagram. Using this fact, we can set up a proportion with one ratio involving segments LD and KM and the other ratio involving segments LT ( we'll let "T" be the point where LD intersects OP ) and OP. If we represent the length of LT as "x", the proportion will look like this: 1/3 = x/OP.

Now we will isolate x in this proportion, so by cross-multiplying, we get 3x = OP, so x = 1/3(OP).

Since we are given that the ratio between NO and OP is 5/9, we now set up another proportion in which we use the ratio between segments TD( whose length can now be represented as 16 - x ) and OP, and set it = to 5/9, so it becomes (16 - x )/OP = 5/9. Since we know that x = 1/3(OP), let's insert this into the proportion, so it now becomes ( 16 - 1/3(OP))/OP = 5/9. Cross-multiplying, we have 144 - 3(OP) = 5(OP). Combining our like terms, we have 144 = 8(OP), so OP = 18 and, by the 5:9 ratio, this forces NO to be 10.