Triangle A has an area of #9 # and two sides of lengths #4 # and #6 #. Triangle B is similar to triangle A and has a side with a length of #16 #. What are the maximum and minimum possible areas of triangle B?

Question

Triangle A has an area of #9    # and two sides of lengths #4  # and #6   #.  Triangle B is similar to triangle A and has a side with a length of #16   #.  What are the maximum and minimum possible areas of triangle B?

Maverick Smith · Accepted Answer

Maximum possible area of triangle B = 144
Minimum possible area of triangle B = 64

#Delta s A and B # are similar. To get the maximum area of #Delta B#, side 25 of #Delta B# should correspond to side 4 of #Delta A#. Sides are in the ratio 16 : 4 Hence the areas will be in the ratio of #16^2 : 4^2 = 256 : 16# Maximum Area of triangle #B =( 9 * 256) / 16= 144# Similarly to get the minimum area, side 6 of #Delta A # will correspond to side 16 of #Delta B#. Sides are in the ratio # 16 : 6# and areas #256 : 36# Minimum area of #Delta B = (9*256)/36= 64#

Wyatt Anderson · Answer

undefined Since triangles A and B are similar, their corresponding sides are proportional.

Let's denote the area of triangle B as $ A_B $. If the side of triangle B that corresponds to the side of length 4 in triangle A is 16, then the scale factor between the two triangles is $ \frac{16}{4} = 4 $.

Area is proportional to the square of the side length in similar triangles. So, if the side length of triangle B is 16 times the side length of triangle A, the area of triangle B will be $ 16^2 = 256 $ times the area of triangle A.

$$ A_B = 256 \times 9 = 2304 $$

So, the maximum possible area of triangle B is $ 2304 $.

For the minimum possible area, consider if the side of triangle B corresponding to the side of length 6 in triangle A is also 16. In this case, the scale factor would be $ \frac{16}{6} = \frac{8}{3} $.

Since area is proportional to the square of the side length, the area of triangle B would be $ \left(\frac{8}{3}\right)^2 = \frac{64}{9} $ times the area of triangle A.

$$ A_B = \frac{64}{9} \times 9 = 64 $$

So, the minimum possible area of triangle B is $ 64 $.