Triangle A has an area of #5 # and two sides of lengths #4 # and #7 #. Triangle B is similar to triangle A and has a side of length #15 #. What are the maximum and minimum possible areas of triangle B?

Question

Triangle A has an area of #5    # and two sides of lengths #4  # and #7   #.  Triangle B is similar to triangle A and has a side of length #15   #.  What are the maximum and minimum possible areas of triangle B?

Violet Aldrich · Accepted Answer

Maximum possible area of triangle B = 70.3125
Minimum possible area of triangle B = 22.9592

#Delta s A and B # are similar. To get the maximum area of #Delta B#, side 15 of #Delta B# should correspond to side 4 of #Delta A#. Sides are in the ratio 15 : 4 Hence the areas will be in the ratio of #15^2 : 4^2 = 225 : 16# Maximum Area of triangle #B =( 5 * 225) / 16= 70.3125# Similarly to get the minimum area, side 7 of #Delta A # will correspond to side 15 of #Delta B#. Sides are in the ratio # 15 : 7# and areas #225 : 49# Minimum area of #Delta B = (5*225)/49= 22.9592#

Autumn Armstrong · Answer

undefined To find the maximum and minimum possible areas of triangle B, we first need to determine the scale factor by comparing the corresponding sides of triangles A and B since they are similar.

The scale factor $ k $ can be found by dividing the length of the corresponding side of triangle B by the length of the corresponding side of triangle A. In this case, the corresponding side of triangle B is 15 and the corresponding side of triangle A is 7. So, the scale factor $ k $ is:

$$ k = \frac{15}{7} $$

Now, to find the maximum and minimum possible areas of triangle B, we'll use the fact that the area of a triangle is proportional to the square of the length of its sides. So, if we increase or decrease the lengths of the sides of triangle B by the scale factor $ k $, the area will be affected accordingly.

Maximum possible area of triangle B:
$$ (\text{Area of triangle A}) \times k^2 $$

Minimum possible area of triangle B:
$$ (\text{Area of triangle A}) \times \left(\frac{1}{k}\right)^2 $$

Substitute the given values into the formulas to find the maximum and minimum possible areas:

Maximum possible area of triangle B:
$$ 5 \times \left(\frac{15}{7}\right)^2 $$

Minimum possible area of triangle B:
$$ 5 \times \left(\frac{7}{15}\right)^2 $$

Calculating these expressions will give us the maximum and minimum possible areas of triangle B.