Lois C. answered • 06/07/20
BA in secondary math ed with 20+ years of classroom experience
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.
Lois C.
Because of the similarity of the triangles, and because LD is perpendicular to KM, the ratio of height to base of the large triangle ( i.e. 16 to 48 ) will be maintained when a segment connects two sides of a triangle ( in this case OP) and runs parallel to the 3rd side ( which we know it does because of the right angles), splitting the original triangle so that a smaller, similar triangle is formed at the top ( in this case, triangle LOP). Hope this helps!06/07/20
Mark M.
What I do not understand is how the ratio can be determined from LD that is the height with KM that is the base. By similarity the ratio would be determined by corresponding parts, e.g., OP and KM. No part of the figure corresponds to LD - an altitude.06/08/20
Lois C.
Since LD is perpendicular to KM and since NOPS is a rectangle, then we know that LT ( see my note on which point I'm calling "T") would also be perpendicular to OP and, by AA similarity theorem, the smaller triangle on top ( triangle LOP ) would be similar to the original larger triangle LKM. The angles that allow for the use of the AA ~ similarity theorem would be angles OLP and KLM and angles LOP and LKM ( or, you could also use the 3rd pair of angles, LPO and LMK). Does this help?06/08/20
Mark M.
how did you establish the ratio of 16/48 or 1:3?06/07/20