Triangle A has an area of #12 # and two sides of lengths #5 # and #7 #. Triangle B is similar to triangle A and has a side with a length of #19 #. What are the maximum and minimum possible areas of triangle B?

Question

Triangle A has an area of #12    # and two sides of lengths #5  # and #7   #.  Triangle B is similar to triangle A and has a side with a length of #19   #.  What are the maximum and minimum possible areas of triangle B?

Sadie Adams · Accepted Answer

Maximum Area #=187.947" "#square units
Minimum Area #=88.4082" "#square units

The triangles A and B are similar. By ratio and proportion method of solution, triangle B has three possible triangles. For Triangle A: the sides are #x=7#, #y=5#, #z=4.800941906394#,Angle #Z=43.29180759327^@# The angle Z between sides x and y was obtained using the formula for area of triangle #Area=1/2*x*y*sin Z# #12=1/2*7*5*sin Z# #Z=43.29180759327^@# Three possible triangles for Triangle B: the sides are Triangle 1. #x_1=19#, #y_1=95/7#,#z_1=13.031128031641#, Angle #Z_1=43.29180759327^@# Triangle 2. #x_2=133/5#,#y_2=19#, #z_2=18.243579244297#, Angle #Z_2=43.29180759327^@# Triangle 3. #x_3=27.702897180004#, #y_3=19.787783700002#, Angle #Z_3=43.29180759327^@# Maximum Area with Triangle 3. Minimum Area with Triangle 1. God bless....I hope the explanation is useful.

Sadie Adams · Answer

undefined To find the maximum and minimum possible areas of triangle B, we first need to determine the scale factor between triangles A and B, as they are similar triangles. Since one side of triangle B is 19 units long and corresponds to a side length of 7 units in triangle A, the scale factor between them is $ \frac{19}{7} $.

To find the maximum possible area of triangle B, we use the fact that the area of a similar triangle is proportional to the square of the scale factor. So, if the area of triangle A is 12 units$^2$, then the maximum possible area of triangle B is $ 12 \times \left( \frac{19}{7} \right)^2 $.

To find the minimum possible area of triangle B, we again use the fact that the area of a similar triangle is proportional to the square of the scale factor. So, the minimum possible area of triangle B occurs when the scale factor is minimized. Since the side length of 19 in triangle B corresponds to a side length of 5 in triangle A, the scale factor is $ \frac{19}{5} $. Therefore, the minimum possible area of triangle B is $ 12 \times \left( \frac{19}{5} \right)^2 $.

Calculating these values, we find:

Maximum possible area of triangle B = $ 12 \times \left( \frac{19}{7} \right)^2 $

Minimum possible area of triangle B = $ 12 \times \left( \frac{19}{5} \right)^2 $

These values can be computed to obtain the numerical results for the maximum and minimum possible areas of triangle B.