Question 1155824: Two similar cones have radii in the ratio n:m. What is the ratio of their slant heights? Found 2 solutions by jim_thompson5910, greenestamps:Answer by jim_thompson5910(35256) (Show Source):
Their slant heights must have the same ratio as the radii in order for the overall cones themselves to be similar. If the slant heights were not in the same ratio as the radii, then we wouldn't have similar cones.
This applies to the heights as well. Let's say they are h1 and h2. So h1:h2 = n:m also. You can use the Pythagorean theorem to find the slant heights. In effect, you'll have two similar right triangles (which are half of a vertical cross section). Then you would use the SSS (side side side) similarity theorem which says that if you have three pairs of sides in the same ratio or proportion, then the two triangles must be similar.
It's a basic principle for similar figures, no matter how simple or complex they are. It can be two squares (which are all similar to each other), or it can be an entire city and a scale model of that city.
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If two similar figures have a scale factor (ratio of corresponding linear measurements) of a:b, then
(1) the ratio of corresponding area measurements is a^2:b^2; and
(2) the ratio of corresponding volume measurements is a^3:b^3
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Radii and slant heights are both linear measurements; the ratio of the two slant heights is the same as the ratio of the radii.
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