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kondaur [170]
3 years ago
10

(Urgent) Worth 15 points 7.3

Mathematics
1 answer:
Stella [2.4K]3 years ago
5 0

Answer: 20790

Step-by-step explanation:

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Find the area of the triangle with sides 40, 50, and 60.
Katyanochek1 [597]

Answer:

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

4 0
2 years ago
Read 2 more answers
A right triangle has side lengths 4 units, 5 units, and x units what is the difference between the difference values of x?
irina [24]

Use the Pythagorean theorem.

If x is a hypotenuse, then we have:

x^2=4^2+5^2\\\\x^2=16+25\\\\x^2=41\to x=\sqrt{41}

If x is a leg, then we have:

x^2+4^2=5^2\\\\x^2+16=25\ \ \ \ |-16\\\\x^2=9\to x=\sqrt9\to x=3

Answer: \sqrt{41}-3

6 0
3 years ago
Which of the following is INCORRECT? *
DedPeter [7]

this option is incorrect

A straight angle is one full revolution.

Step-by-step explanation:

Straight angle = 180°

Angle take 360° in one revolution

6 0
2 years ago
12 stamps increased by 125% what is the new amount
GREYUIT [131]

12 x 1.25 = 15

15 is the new amount

4 0
3 years ago
Convert 9 over 16 to a decimal.<br> A. 0.4225<br> B. 0.5225<br> C. 0.5625 <br> D. 0.7455
Brums [2.3K]
C. 0.5625...you find the answer by dividing 6 by 19 :)
7 0
2 years ago
Read 2 more answers
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