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Question 96231: Help again! It seems there are different ways to figure out problems for rectangles, squares, and triangles. Is there a different formula for all of them. The questions I'm working with give me the perimeter I have to figure out the dimensions. I FEEL SO OVERWHELMED!
ex. A square has a perimeter of 28 feet. Find the length of each side.
ex. A triangle has a perimeter of 23 inches. The medium side is 3 inches more than the shortest side, and the longest side is twice the shortest side. Find the shortest side.
Found 2 solutions by bucky, stanbon:
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Whenever you feel overwhelmed, you should always try to go back to the basic things you
have learned. For example, these two problems involve perimeters. You should know that, by
definition, the perimeter of a geometric figure is just the sum of the lengths of all the sides.
.
The first problem involves the perimeter and a square. What do you know about squares? A square
is a special case of a rectangle. It has four sides and 4 angles of 90 degrees. In addition,
what makes a square different from most rectangles? You probably know that all the sides
of a square are equal. Let's use this basic information to solve the first problem.
.
Since the perimeter of a four sided figure is the sum of the lengths of the four sides, we
can write an equation:
.
P = a + b + c + d
.
in which P represents the perimeter and a, b, c, and d represent the 4 sides. But in
a square, all the sides are equal in length. Let's call that length "a". This makes
the perimeter equation become:
.
P = a + a + a + a
.
and the four terms on the right side add up to 4a.
.
Therefore, for a square we can say
.
P = 4a
.
where "a" is the length of one of the sides. The problem tells you that the perimeter of
the square is 28 feet. So we can replace P with 28 in the equation to get:
.
28 = 4a
.
Finally, we can solve for "a" (which is the length of one side) by dividing both sides of
this equation by 4. This division will make the right side of the equation become just
"a" and the left side of this equation will be 28/4 which is 7. So the equation reduces to:
.
7 = a
.
And that is the answer. The length of any side of the square is 7 feet.
.
Now on to the second problem. This problem involves a triangle. And we know that a triangle has
three sides. A triangle can have all sides equal, two sides equal, or no sides equal. In
this problem the sides are described as shortest, medium, and longest. So we have a
triangle with all sides different in length. Since we know that the triangle has 3 sides
and all sides are different we can write the perimeter equation as:
.
P = a + b + c
.
where P is the perimeter and a, b, and c are the different lengths of the 3 sides.
.
Now let's look at the information in the problem about the three sides. Let's call the
shortest side "a". Then note that the other sides (b and c) are expressed in terms of
"a". The medium side (call it side b) is 3 inches longer than "a". So side b is equal to a + 3.
Then we are told that the longest side (side c) is twice as long as the shortest side "a". So
we know that c = 2a.
.
Now let's return to the perimeter equation and substitute "a" for "a", "a + 3" for b, and
2a for c. When we do those substitutions the perimeter equation becomes:
.
P = a + a + 3 + 2a
.
If we add up all the terms on the right side that contain the letter "a" the equation
becomes:
.
P = 4a + 3
.
Now, recall that the problem told us that P was 23 inches. So, let's substitute 23 inches
for P and get:
.
23 = 4a + 3
.
We need to get rid of the +3 on the right side. Do this by subtracting 3 from both sides
and this makes the equation become:
.
20 = 4a
.
Solve for "a" (the shortest side) by dividing both sides by 4 and you have:
.
5 = a
.
So "a" is 5 inches. The medium side is 3 inches longer than the short side. So the medium
side is 5 + 3 = 8 inches. And the longest side is twice the length of the shortest side.
So the longest side is 2 times 5 which equals 10 inches.
.
In summary the three sides are 5, 8, and 10 inches.
.
To check, add these three sides together and you should have the perimeter. Sure enough ...
5 + 8 + 10 does equal 23 inches, just as the problem said it should.
.
Hope this helps you to understand the problems and how to solve them. Don't panic about
this stuff ... stay cool and go back to the basics. You don't even have to say what equation
should I use ... just use what you already know and try to think your way through the
problem. That's what math is all about. Eventually, it will get to be second nature to you
... once you get a lot of practice in. Good luck and keep plugging away at it.
.
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website! ex. A square has a perimeter of 28 feet. Find the length of each side.
1st: Draw a picture of a square (rectangle with all sides equal)
Label each side "x".
EQUATION:
Perimeter is the distance around the square:
4x = 28
Divide both sides by 4 to get:
x = 7 ft (lenght of each of the sides)
==========================
ex. A triangle has a perimeter of 23 inches. The medium side is 3 inches more than the shortest side, and the longest side is twice the shortest side. Find the shortest side.
__________________
Draw a triangle.
Label the shortest sides "x".
Label the longest side "2x"
Label the remaining side "x+3".
-------------------
EQUATION:
The perimeter is the sum of the sides:
x + 2x + x+3 = 23
Simplify the equation:
4x+3 = 23
Subtract 3 from each side to get:
4x = 20
Divide both sides by 4 to get:
x = 5 (length of shortest side)
x+3 = 5+3 = 8 (length of 2nd side)
2x = 2*5 = 10 (length of the longest side)
===================
Cheers,
Stan H.
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