Triangle A has an area of #6 # and two sides of lengths #9 # and #4 #. Triangle B is similar to triangle A and has a side with a length of #14 #. What are the maximum and minimum possible areas of triangle B?

Question

Triangle A has an area of #6    # and two sides of lengths #9  # and #4   #.  Triangle B is similar to triangle A and has a side with a length of #14   #.  What are the maximum and minimum possible areas of triangle B?

Aaliyah Anderson · Accepted Answer

Possible maximum area of triangle B = 73.5
Possible minimum area of triangle B = 14.5185

#Delta s A and B # are similar. To get the maximum area of #Delta B#, side 14 of #Delta B# should correspond to side 4 of #Delta A#. Sides are in the ratio 14 : 4 Hence the areas will be in the ratio of #14^2 : 4^2 = 196 : 16# Maximum Area of triangle #B =( 6 * 196) / 16= 73.5# Similarly to get the minimum area, side 9 of #Delta A # will correspond to side 14 of #Delta B#. Sides are in the ratio # 14 : 9# and areas #196 : 81# Minimum area of #Delta B = (6*196)/81= 14.5185#

Noah Atkinson · Answer

undefined The maximum possible area of triangle B occurs when it is similar to triangle A and its corresponding sides are proportional to the corresponding sides of triangle A. Since triangle A has an area of 6 and a base of length 9, its height (corresponding to the side of length 4) is 6/2 = 3. Thus, the ratio of corresponding sides of triangle B to triangle A is 14/9.

The area of triangle B can be calculated using the formula for the area of a triangle: Area = (1/2) * base * height. Since the base of triangle B is 14 and the corresponding height (proportional to the height of triangle A) is (14/9) * 3 = 14/3, the maximum possible area of triangle B is (1/2) * 14 * (14/3) = 98/3 or approximately 32.67 square units.

The minimum possible area of triangle B occurs when it is still similar to triangle A but is scaled down as much as possible while maintaining the proportions of the sides. In this case, the height of triangle B would be 3/4 times the height of triangle A (since the side of length 14 is 3.5 times longer than the side of length 4 in triangle A). So, the minimum possible area of triangle B is (1/2) * 14 * (3/4 * 3) = 63/4 or approximately 15.75 square units.