Question 321627: The question states which of the 2 triangles has the larger area? One triangle has a base of 8 and 2 angles of 5 each. The second triangle has a base of 6 and the other 2 sides are both 5. According to the book, these are supposed to be equal. I don't understand what I did wrong but I got that the first one is larger. How do I do this? Thanks.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Looks like these are both isosceles triangles.
The second triangle has a base of 6 and 2 sides of 5.
The height of this triangle has to be 4.
Since this triangle is isosceles, drop a perpendicular from the angle opposite the base to get 2 right triangle of height x and base 3 and hypotenuse of 5.
Since this is a 3/4/5 triangle, the height has to be 4.
You can also determine the base angle by taking the arc cosine of 3/5 = 53.13010235 degrees.
tan of 53.12010235 = x/3 which meakes x = 3 * tan(53...) = 4
Area of this triangle = 1/2 * base * height = 1/2 * 6 * 4 = 1/2 * 24 = 12.
Unless I read this wrong, the second triangle can't possibly have the same area.
The base is 8.
Each of the base angles are 5 degrees apiece???????
Once again this is an isosceles triangle.
Drop a perpendicular from the angle opposite the base and you get two right triangles with a base of 4 each and an angle of 5 degrees with another angle of 85 degrees.
Since tan(5) = altitude / 4, this makes altitude = 4 * tan(5) = 4 * .087488664 = .349954654.
area of the triangle is 1/2 * b * h = 1/2 * 8 * .349954654 = 1.299818616.
The area of this triangle is much smaller than the area of the other triangle.
I don't know how they could possible be equal unless you got the degrees wrong.
For this triangle to have the same area, then 12 = 1/2 * b * h = 1/2 * 8 * h = 4 * h which means that h had to be equal to 3 (height / altitude).
For that to happen, the angles had to be 36.86989765 degrees.
If you were talking the size of the sides rather than the sides of the angle, then this makes more sense because you have 2 triangles.
One triangle has a base of 6 and 2 sides of 5.
The other triangle has a base of 8 and 2 sides of 5.
The first triangle has an altitude of 4.
The second triangle has an altitude of 3.
Both these triangle are isosceles so you can drop a perpendicular from the angle opposite the base to the base.
The first triangle becomes 2 right triangles of dimensions 3,4,5 (base,height,hypotenuse).
The second triangle becomes 2 right triangles of dimensions 4,3,5 (base,height,hypotenuse).
A right triangle that's 3,4,5 has the same area as a right triangle that is 4,3,5 and so the areas of these right triangles are equal which makes the areas of the isosceles triangles they are part of equal as well.
I'll bet that's what they meant.
Both sides of the triangles are 5 in length.
Angle of 5 doesn't apply.
Check the question again to see if this makes more sense to you.
Area of the triangle is 1/2 * b * h.
1/2 * 4 * 6 = 2 * 6 = 12
1/2 * 3 * 8 = 3 * 4 = 12
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