Question - Solving Triangles with Given Ratios
Solution:
The image shows two triangles, with a given statement that "The ratio of the LARGER triangle to the SMALLER triangle is $$ \frac{3}{5} $$."To use this information to solve the problem, we must understand that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. In this case, the larger triangle has sides of lengths 21 and 18, but we are not given the corresponding sides of the smaller triangle. However, we do know one of the sides of the smaller triangle, which is 5.Since we know the ratio of the areas is $$ \frac{3}{5} $$, we can set up an equation using the side of the smaller triangle that we know:\[ \left( \frac{\text{Side of larger triangle}}{\text{Side of smaller triangle}} \right)^2 = \frac{3}{5} \]We can use either 21 (corresponding to the larger side) or 18 (corresponding to the smaller side) for the "Side of larger triangle" in the equation. Let's use 18 as it corresponds with the side length of 5 in the smaller triangle:\[ \left( \frac{18}{5} \right)^2 = \frac{3}{5} \]To find the "Side of larger triangle" that corresponds to the smaller triangle's side of length 5, we can use the square root to get:\[ \left( \frac{18}{5} \right) = \sqrt{\frac{3}{5}} \]\[ \frac{18}{5} = \frac{\sqrt{3}}{\sqrt{5}} \]Now, to solve for the corresponding side in the larger triangle, we cross-multiply:\[ 5\sqrt{3} = 18\sqrt{5} \]\[ \text{Side of larger triangle} = \frac{18\sqrt{5}}{\sqrt{3}} \]\[ \text{Side of larger triangle} = \frac{18\sqrt{3}\sqrt{5}}{3} \]\[ \text{Side of larger triangle} = 6\sqrt{15} \]So the side of the larger triangle that corresponds to the side of length 5 in the smaller triangle has a length of $$ 6\sqrt{15} $$, which would be the correct value to fill in the blank space in the image.
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