Triangle A has an area of #13 # and two sides of lengths #2 # and #14 #. Triangle B is similar to triangle A and has a side of length #18 #. What are the maximum and minimum possible areas of triangle B?

Question

Triangle A has an area of #13    # and two sides of lengths #2  # and #14   #.  Triangle B is similar to triangle A and has a side of length #18   #.  What are the maximum and minimum possible areas of triangle B?

Aaliyah Anderson · Accepted Answer

Maximum possible area of triangle B = 1053
Minimum possible area of triangle B = 21.4898

#Delta s A and B # are similar. To get the maximum area of #Delta B#, side 18 of #Delta B# should correspond to side 12 of #Delta A#. Sides are in the ratio 18 : 2 Hence the areas will be in the ratio of #18^2 : 2^2 = 324 : 4# Maximum Area of triangle #B =( 13 * 324) / 4= 1053# Similarly to get the minimum area, side 14 of #Delta A # will correspond to side 18 of #Delta B#. Sides are in the ratio # 18 : 14# and areas #324 : 196# Minimum area of #Delta B = (13*324)/196= 21.4898#

Sophia Alfred · Answer

undefined To find the maximum and minimum possible areas of triangle B, which is similar to triangle A and has a side of length 18, we can use the properties of similar triangles.

Since triangle B is similar to triangle A, their corresponding sides are in proportion. Let $ k $ be the scale factor of similarity.

The side lengths of triangle A are given as 2 and 14. The corresponding side lengths of triangle B, which has a side of length 18, would be $ 2k $ and $ 14k $.

The area of a triangle can be calculated using the formula $ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $.

For triangle A, the base and height can be any combination of its sides. Let's consider the base as 14 and the height as the corresponding height for this base. Therefore, the height of triangle A can be calculated using the formula:

$$ \text{Area of triangle A} = \frac{1}{2} \times 14 \times \text{height of triangle A} $$

Solving for the height of triangle A, we find:

$$ \text{Height of triangle A} = \frac{2 \times \text{Area of triangle A}}{14} = \frac{2 \times 13}{14} $$

Now, using the scale factor $ k $, the corresponding height for triangle B would be $ \frac{2 \times 13k}{14k} = \frac{13}{7} $.

Now, to find the maximum and minimum possible areas of triangle B, we consider different combinations of base and height for triangle B.

The maximum area would occur when the base is 18 and the height is $ \frac{13}{7} $, resulting in an area of:

$$ \text{Maximum Area of triangle B} = \frac{1}{2} \times 18 \times \frac{13}{7} $$

The minimum area would occur when the base is $ 2k $ and the height is $ \frac{13}{7} $, resulting in an area of:

$$ \text{Minimum Area of triangle B} = \frac{1}{2} \times 2k \times \frac{13}{7} $$

Therefore, the maximum possible area of triangle B is $ \frac{1}{2} \times 18 \times \frac{13}{7} $ and the minimum possible area of triangle B is $ \frac{1}{2} \times 2k \times \frac{13}{7} $.