Triangle A has an area of #6 # and two sides of lengths #3 # and #8 #. Triangle B is similar to triangle A and has a side with a length of #7 #. What are the maximum and minimum possible areas of triangle B?

Answer 1

#"Area"_"Max" = 10.5#

#"Area"_"Min" = 1.43#

From the sides and area, we calculate the remaining side of the first triangle as . #A = 1/2b xx h# with 8 as b, #6 = 4 xx h#; #h = 1.5# Separating the 8 side into x and 8 - x we obtain: #3^2 = 1.5^2 + x^2# and #c^2 = 1.5^2 + (8-x)^2# #9 = 2.25 + x^2# ; #x^2 = 6.75#; #x = 2.6# #8-x = 5.40# #c^2 = 1.5^2 + (8-x)^2#; #c^2 = 2.25 + 5.4^2# #c = 5.60#
The maximum similar triangle 'B' would have 7 as the shortest side, and the minimum would have it as the longest side. Those triangles would have sides of 7, 13.1, 18.7 and 7, 4.90, 2.63 respectively. The ratio of the base split (#2.6/8#) applied to each gives us: #"Base"_"Max" = 0.325 xx 18.7 = 6.08 #(h_"Max")^2 = 7^2 - 6.08^2#; #h_"Max" = 3.47#
#"Base"_"Min" = 2.275# (derived similar to previous) #(h_"Min")^2 = 2.6^2 - 2.275^2#; #h_"Max" = 1.26#
#"Area"_"Max" = 1/2 xx 6.08 xx 3.47 = 10.5#
#"Area"_"Min" = 1/2 xx 2.275 xx 1.26 = 1.43#
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Answer 2

The maximum possible area of triangle B is ( \left(\frac{7}{8}\right)^2 \times 6 = \frac{147}{16} ) square units.

The minimum possible area of triangle B is ( \left(\frac{7}{3}\right)^2 \times 6 = \frac{294}{3} ) square units.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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