Find the area of the triangle with sides 40, 50, and 60. a. 75 square units
b. 992.2 square units
c. 3000 square units
d. 114.6 square units

B. 992.2

Step-by-step explanation:

Area of Triangle Knowing All Sides:

Sides: 50.000, 60.000, 40.000

Area  : 992.1567

Area of Triangle given by its 3 Sides:

I will show two ways to find the area. One way is very short - The 2000 years old Heron's Formula. The other method may be longer but it's more "educational" as it teaches us important analytic geometry lessons.

But before we even start, we have to verify that the Basic Triangle Inequality is satisfied.

The Basic Triangle Inequality:

The Basic Triangle Inequality states that for a triangle with side lengths a,b,c , the following is true:

    a < b + c,    b < a + c,    c < a + c

What this means, in plain English, is that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Let us now check:

     60.000 < 40.000 + 50.000   true

     40.000 < 60.000 + 50.000   true

     50.000 < 60.000 + 40.000   true

Now that the basic inequalities are satisfied, we know that these three side lengths can make a triangle, so we can move on to calculating the area of said triangle

Heron's Formula for the area of a triangle:

The 2000 year old Heron's Formula states that the area of a triangle whose sides have lengths  a, b,  and  c  is

     SQRT (s(s-a)(s-b)(s-c))

where  s  is the semiperimeter of the triangle; that is,

     s = ( a + b + c ) / 2

Let us calculate

Area =  SQRT ( 75.000 • 15.000 • 35.000 • 25.000 ) =

           SQRT ( 984375.000 ) =

           992.1567  

Find area using the Base Height formula:

Let  B  (for base) denote the length of the longest side of the triangle

Let  h  (for height) denote the length of a perpendicular line, from the vertex opposite that side, to the side itself.

Note that  h  splits our triangle into two right-angled triangles, both having  h  as height, the base of the left triangle is denoted by  X  and the base of the right triangle is denoted by  40.000 - X

To find the area of a triangle, multiply the base by the height, and then divide by 2. In algebraic notation, Area  = 0.5 • 40.000 • h   To be able to find the Area, using this formula, we must know the value of  h

Using the Pythagorean theorem to find the Height:

Applying the Pythagorean theorem to the left right-angled triangle we get:

     h2 = (50.000)2- X2  

While the right right-angled triangle is "telling" us that:

     h2 = (60.000)2- (40.000 - X)2  

Two things which are equal to  h2 , are also equal to one another (It is a property of "Equation" that   if z=p and z=q then p=q ):

     (50.000)2- X2 = (60.000)2- (40.000 - X)2  

Expand the above and simplify :

     (40.000)2 + (50.000)2 - (60.000)2 = 80.000 • X

     500.000 = 80.000 • X

     X = 6.250

Plug this for X in:  h2 = (50.000)2- X2  

     h2 = (50.000)2- ( 6.250)2  

     h2 = (2460.938)

h = sqrt (2460.938) = 49.6078  

Put the Triangle Area Formula to use:

Finally put the formula  Area = Base * Height * 0.5   to use:

    Area  = 40.000 • 49.608 • 0.5

    Area  = 992.1567  

Note that this result is identical to the one we got using Heron's Formula !

Area of Triangle Knowing all Sides :

    Sides: 50.000, 60.000, 40.000      Area  : 992.1567

  b.   the average rate of change of f(x) is less than the average rate of change of g(x).

step-by-step explanation:

since the length of the interval is the same in each case, we only need to compare the change in y-value to see which is greater.

so the change in y-value for function f is (6 )) = 9.

the table tells us

so the change in y-value for function g is (31 -1) = 30.

the change in y-value of f(x) on the interval is less than the change in y-value of g(x) on the same interval, so the average rate of change of f(x) is less than the average rate of change of g(x).