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Find the area of triangle \(ABC\) if \(AB = 6\)  , \(BC = 8\) and \(∠ABC = 135°\)\(\)

 Jun 23, 2023
 #1
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I've drawn a diagram here https://www.simpleimageresizer.com/_uploads/photos/7aca4b83/20230623_125748_29.jpg
 

I don't understand how I am supposed to find the altitude of the triangle...this isn't a right triangle and there are no similar triangles visible in the diagram...so I can't use the Pythagorean theorem,...so on my first try, I've tried finding all the angles inside the triangles here...and I've tried using basic trigonometry to help me find one of the side lengths...but that didn't really help me with finding the altitude...

I really need help and fast!

 Jun 23, 2023
 #2
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The area of a triangle can be calculated using the following formula:

Area = (1/2) * base * height

In this case, we know that the base of the triangle is BC = 8, and the height of the triangle is the perpendicular distance from A to BC.

We can use the sine function to calculate the height of the triangle:

sin(angle ABC) = height / BC

sin(135 degrees) = height / 8

(1/2) = height / 8

height = 4

Now that we know the base and height of the triangle, we can calculate the area:

Area = (1/2) * 8 * 4 = 16

 

Therefore, the area of triangle ABC is 16.

 Jun 23, 2023
 #8
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I think you are on the right track, but \(\sin 135^\circ \neq \frac{1}{2}\)

Guest Jun 23, 2023
 #3
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I've tried entering that answer,...but it's wrong...thanks for trying though!

 Jun 23, 2023
edited by icecreamlover  Jun 23, 2023
 #4
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I still need help...

will anyone please help me?

 Jun 23, 2023
 #5
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Here is my attempt at solving this.

 

I would solve this problem by rotating the triangle. Consider the following diagram: 

 

Now, notice how the height in this diagram is side length EB. The given info says that BC = 8, Since triangle BEC is a 45-45-90 triangle, we can determine that the height of the triangle is sqrt(18) or 3 * sqrt(2). We know that AB = 8, so we can find the area using the basic formula for the area of a triangle, Area = 1/2 * base * height.

 

Area = 1/2 * 8 * 3 * sqrt(2) = 12 * sqrt(2), so I believe this is the right answer.

 Jun 23, 2023
 #6
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The area of a triangle can be calculated using the following formula:

Area = (1/2) * base * height

In this case, we know that the base is BC = 8 and the angle between AB and BC is 135 degrees.

The height of the triangle is equal to the length of the altitude from A to BC. We can find the length of the altitude using the following formula:

altitude = AB * sin(angle ABC)

Plugging in the values we know, we get:

altitude = 6 * sin(135 degrees) = 6 * (1/2) = 3

Therefore, the area of triangle ABC is:

Area = (1/2) * 8 * 3 = 12

So the answer is 12​.

 Jun 23, 2023
 #7
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I think you are on the right track, but you made a miscalculation. \(\sin 135^\circ \neq \frac{1}{2}\)

Guest Jun 23, 2023
 #9
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I think the answer is 12 sqrt2... I tried the formula Guest described and that's what I got...

HumenBeing  Jun 24, 2023
edited by HumenBeing  Jun 24, 2023
 #10
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Explanation: if one side of that 45-45-90 is 6, the other 2 congruent sides must be 6/sqrt2... which equals 3sqrt2. and multiply that by 8 and divide by 2, and you get 12sqrt2. 

 Jun 24, 2023
 #11
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thanks, but none of these answers were correct...

 Jun 25, 2023
 #12
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can anyone else please help?

 Jun 25, 2023
 #13
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anyone??

 Jun 25, 2023
 #14
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nvm, I found the answer

 Jun 25, 2023

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