Find the area & perimeter of the figure

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Find the area & perimeter of the figure

Geometry
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1 Attachment
You were given a triangle like this: |dw:1437006808180:dw|
What do the circled marks mean? |dw:1437006848302:dw|

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@11.seventeen what do the circled marks mean
doesn't it mean its equal
Yes. All sides are congruent. That means every side of this triangle measures 2a mm. That makes the perimeter easy to find. What is the perimeter of a triangle?
\(\Large P_{triangle} = a + b + c\) where a, b, and c are the lengths of the sides of the triangle.
\(P = 2a + 2a + 2a = 6a\) The perimeter of the triangle is 6a mm
Ok so far?
Yes. It's the same as what i got @mathstudent55
Now you have to find the area.
And I would do that how ? Because I don't have the heigh . I have the base is 2
To find the area, you need to find the height of the triangle since the formula for the area involves the height.
Have you heard about the ratios of the lengths of the sides of a 30-60-90 triangle?
Yes but I never understood it
Ok. I'll explain it to you. Here is a triangle with a right angle and a 30-deg angle and a 60-deg angle. |dw:1437011007751:dw|
Since the right angle is 90 deg, this triangles' three angles have measures 30, 60, and 90 degrees. This is what is called a 30-60-90 triangle.
In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the short leg. If you call the length of the short leg 1, then the hypotenuse has length 2. |dw:1437011223749:dw|
The length of the long leg is \(\sqrt 3\) times the length of the short leg.
|dw:1437011291789:dw|
So far all we have is that the lengths of the sides of a 30-60-90 triangle are in the ratio of 1 : \(\sqrt 3 : 2\)
Do you understand so far?
sort of
It is simpler than you think. Think of \(\sqrt 3\) as being approximately 1.7 All this means is that: In a 30-60-90 triangle, the long leg is 1.7 times the length of the short leg. The hypotenuse is 2 times the length of the short leg.
If a 30-60-90 triangle has a short leg of 1, then the long leg is 1.7 * 1 = 1.7 the hypotenuse is 2 * 1 = 2
how do I get the height out of this?
Another example: If a 30-60-90 triangle has a short leg of length 4, then the long leg is 1.7 * 4 = 6.8 and the hypotenuse is 2 * 4 = 8
Ok. Let's get back to our problem.
|dw:1437011694076:dw|
You see our triangle above. Since all sides are congruent, all angles are also congruent. That means all angles measure 60 degrees. Ok?
|dw:1437011786387:dw|
We are looking for the length of the height. By drawing the height, we created two new triangles. Notice each small triangle is a 30-60-90 triangle.
|dw:1437011853170:dw|
Look at the figure above. Each of the two small triangles is a 30-60-90 triangle. The 30 degree angle is above. Since the side of the large triangle has length 2a, half of that is just a, and that is the length of the short side of the 30-60-90 triangle.
Look at the small triangle in the circle and ignore the rest. |dw:1437012005959:dw|
The short leg has length a. The long leg has length \(\sqrt 3\) times a, which is simply: \(\sqrt 3 a\) |dw:1437012059638:dw|
|dw:1437012157126:dw|
The long leg of the small triangle is the height of the large triangle. That was the height we needed for the area.
|dw:1437012211126:dw|
so is the height 3a?
Now we have all the info we need to find the area of the large triangle.
The height is \(\sqrt 3 a\), not 3a. \(\Large A_{triangle} = \dfrac{bh}{2} \)
\(\Large A = \dfrac{(a)(\sqrt 3 a)}{2} = \dfrac{\sqrt 3}{2}a^2 ~mm^2\)
thanks!!!
You're welcome.

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