+197 Find the area of triangle \(ABC\) if \(AB = 6\) , \(BC = 8\) and \(∠ABC = 135°\)\(\)
+197 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!
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.
+197 I've tried entering that answer,...but it's wrong...thanks for trying though!
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.
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.
I think you are on the right track, but you made a miscalculation. \(\sin 135^\circ \neq \frac{1}{2}\)
+184 I think the answer is 12 sqrt2... I tried the formula Guest described and that's what I got...
+184 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.
