Triangle ABC has an area of 30cm^2, AC = 9cm, BC = 7cm. Calculate the largest possible length of AB?

Answer 1

This was not all that difficult.

I used the basic area of a triangle equation to come up with the altitude of the triangle. A = ½×a×b, where a is the altitude and b is the base (any one side).
A = ½×a×b
30 = ½×a×9
a = 60/9 = 6.6666~

I then had to use the the definition of sines to determine an angle.
sin θ = opposite/hypotenuse = a/7 = 6.6666~/7
θ = arcsin (6.666~/7) = 72.2472°

Then I determined the supplemental angle, since there are two values of angles that produce the same sine value. This is because of ambiguity, look up "ambiguous case."
θ' = 180° - θ = 180° - 72.2472°
θ' =107.7528°

Then I used the law of cosines to determine the unknown side.
AB² = AC² + BC² - 2 × AC × BC × cos θ
AB² = 9² + 7² - 2× 9 × 7 × cos 72.2472°
AB = 9.5698

AB² = AC² + BC² - 2 × AC × BC × cos θ'
AB² = 9² + 7² - 2× 9 × 7 × cos 107.7528°
AB = 12.9776

Obviously, 12.9776 is the longest possible length.

Answer 2

We will using the Pythagoras theorem to determine that.
The formula is A^2 +B^2 = C^2
The C^2 is the hypotenuse, which is the largest side on a right triangle (Hypotenuse is the side opposite to the 2 sides that created the perpendicular angle, in this case, it's AB, the other two sides are legs). The reason I choose AB is the hypotenuse is because we are trying to find the largest possible length of AB and the hypotenuse is the largest side. Okay, let's put everything into the equation.
AC^2 +BC^2 = AB^2
9^2 + 7^2 = AB^2
81 + 49 = AB^2
130 = AB^2
By square root, we got
AB= 11.40 (Usually we shorten to 2 decimal place)
The area is just a distraction from the real answer.
Hope that help.

Answer 3

Triangle ABC where BC = 7 cm, AC = 8 cm and angle ACB = 59° ..... the arc length.
ii. the sector area

Answer 4

you can calculate the area for a triangle with the heron formula wich is
sqrt(p(p-a)(p-b)(p-c))
where p=(a+b+c)/2
and a , b , c are the lengths of AB, AC, BC
if you solve the equation
p=x+7+9
sqrt(p(p-a)(p-b)(p-c))=30
you will find the exact value of x, wich is the length AB

Answer 5

11.4 cm