The measurements of the sides of a triangle are consecutive even integers, and the perimeter of the triangle i

Question

The measurements of the sides of a triangle are consecutive even integers, and the perimeter of the triangle is 72 inches. Find the length of the sides and find the area of the triangle. Show all work.

Answer 1

To find it, I know that there are three sides that will add up to 72 and they are consecutive numbers. I would divide 72 by 3. That gives me 24. Ok, so 23 + 24 + 25 will give 72. There is probably a better way to find this but this is how I do it.

Answer 2

I take it by consecutive even integers you mean something like 2, 4, 6, or 10, 12, 14, right?

If so, then the perimeter can be represented by x + (x+2) + (x+4) = 72. Simplifying, I get 72 = 3x + 6, subtracting 6 from each side yields 66 = 3(x), dividing by 3 gets 22 = x, which means the sides are 22, 24, 26. I leave the rest to you to find the area now that I've solved the side lengths. But I'll get you started. I like to put the biggest side on the bottom, the shortest side on the left and the middle length side on the right. What you are looking for is the altitude. Dropping a vertical from the apex divides the base into 2 sections, x (the left half of the base) and (26-x) with the altitude being a. I know that a^2 + x^2 =22^2 AND a^2 + (26-x)^2=24^2, so now you have two equations in 2 unknowns, which I leave you to solve for the altitude a and x.

Answer 3

Let the measurements be:
n = First even integer
n+2 = Second even integer
n+4 = Third even enteger

Then the perimeter is the sum of the sides:
n + (n + 2) + (n + 4) = 3n + 6

And this is equal to 72:
3n+6 = 72

Subtract 6 from both sides:
3n = 66

Divide both sides by 3:
n = 22

So the lengths are 22, 24 and 26.

To find the area use Heron's formula for a triangle (given just the measurement of the sides).

A = sqrt( s (s-a) (s-b) (s-c) )

In this formula, s is the "semi-perimeter" which is simply half the perimeter:
s = (a+b+c)/2 = 36

A = sqrt( 36 * (36-22) * (36-24) * (36-26) )
A = sqrt( 36 * 14 * 12 * 10 )
A = sqrt( 60480 )
A ≈ 245.9268

So the sides are 22, 24 and 26 inches and the area is approximately 246 sq. inches.

Answer 4

Is it really fair for us to do your homework?...

First, call the sides are a, b, c( c is the hypotneuse).
since b is 2 more than a you can call the sides a, a+2, a+4
since the perimeter is 72 that means:

a + b + c =72 or

a + (a+ 2) + (a+4) = 72 or
3a + 6 = 72 which means
3a = 72 - 6
3a = 66
a = 22 that means

a = 22, b = 24 , c = 26

Answer 5

As you're saying on your notes, deriving the sine or cosine isn't undemanding here, so the formula making use of trignometric purposes are no longer the respond. i'm thinking that formula A = ?(s(s-a)(s-b)(s-c)) does the trick. sure, it produces a level 4 polynomial, however the polynomial is of the form ax^4 + bx² + c, as is shown under. on condition that, you may cut back it to a quadratic via the substitution z = x², turning the polynomial into the form az² + bz + c. As for resolving the formula, assume a and b are the conventional sides, and c is the unknown ingredient. Then: A = ?(s(s-a)(s-b)(s-c)) A² = s(s-a)(s-b)(s-c) on condition that s = (a + b + c)/2: A² = ((a + b + c)(a + b - c)(a - b + c)(b - a + c)/sixteen 16A² = ((a + b + c)(a + b - c) * (a - b + c)(b - a + c) Now, (a + b + c)(a + b - c) could be factored as (a + b)² - c², and (a - b + c)(b - a + c) could be factored as c² - (a-b)². So, we are able to write the final equation as 16A² = ((a + b)² - c²)(c² - (a - b)²) ...and, on an analogous time as the excellent equation looks messy adequate, it relatively is obvious that, while a, b, and A are everyday, you may cut back the equation to a quadratic via the substitution z = c². that provides you the thank you to remedy for c. (in case you think of approximately it, fixing for c isn't all that messy once you plug in values for a, b, and A.)

Answer 6

Let s = smallest side

s + (s+2) + (s+4) = 72

3s = 66
s = 22
other sides 24, 26

You will have to plot this and figure out the height of the triangle, then use Area = 1/2 base*height

Answer 7

n-2 + n + n+2 = 72
3n = 72
n = 24
sides are 22, 24, 26
semiperimeter is 72/2 = 36
Heron's formula says
A = √(36)(36-22)(36-24)(36-26)
A = √(36)(14)(12)(10)
A = √(36)(16)(7)(3)(5)
A = 24√105
A = 245.93 in²