imageshack(.)(us)/photo/my-images/42/2vja.jpg/
Open that link please , just delete the parentheses please
please help me with my assignments
I beg you cooperate with me huhu
1. in the figure , the areas of traingle cef, triangle abe, triangle adf are 3,4, and 5 respectively. find the area of triangle aef
2. equialateral triangle abc has an area of square root of 3 and side of length 2. point p is an arbitrary point in the interioir of the traignle. what is the sum of the distances from p to ab, ab, and bc?
3. in the accompanying firgure , abcd is a square . suppose be=3 cm, ec=1cm and p is a point on the diagonal bd. find the smallest possible value of pe + pc in cm.
4. in triangle abc, ab=7, ac=9. on ab, point d is taken so that bd = 3. de is drawn cutting ac in e so that quadrilateral bced has 5/7 the area of trangile abc. find ce.
5.READ THIS CAREFULLY. find the ratio between the area of a square inscribed in a circle and an equilateral circumscribed about the same circle.
Going back to your previous posts, I did read #5 carefully when you posted these before.
At the end you said "an equilateral circumscribed about the same circle"
I can only interpret that as "an equilateral triangle circumscribed about the same circle", and that is how I did the question.
I will now do #3, where ABCD is a square, and P is any point on the diagonal BD.
Using your diagram and information I placed the square on the x-y grid, with the following points
B(0,0), E(3,0) , C(4,0), D(4,4) and A(0,4)
equation of BD is y = x
So I let P be (x,x)
sum of PE + PC
= √( (x-3)^2 + x^2) + √( (x-4)^2 + x^2)
= (2x^2 -6x+9)^(1/2) + (2x^2 - 8x + 16)^(1/2)
d(sum)/dx = (1/2)(2x^2 - 6x + 9)^(-1/2) (4x-6) + (1/2)(2x^2 - 8x + 16)^(-1/2) (4x-8)
= (2x-3)/√(2x^2 - 6x + 9) + (2x-4)/√(2x^2 - 8x + 16)
= 0 for a max/min of sum
(2x - 3)/√2x^2 - 6x + 9) = (-2x +4)/√)2x^2 - 8x + 16)
square both sides to get
(4x^2 - 12x+9)/(2x^2-6x+9) = (4x^2 - 16x + 16)/(2x^2 - 8x + 16)
cross-multiplying .....
8x^4 - 56x^3 + 178x^2 - 264x + 144 = 8x^4 -56x^3+164x^2-240x+144
which surprisingly simplifies to
14x^2 - 24x = 0
7x^2 - 12x = 0
x(7x-12) = 0
x = 0 or x = 12/7
if x = 0 , sum = √9 +√16 = 7 , ( P is at origin)
if x = 12/7 ,
..... I used by calculator to get
sum = 5 (exactly)
(There is probably a much simpler way of doing it, and I think my solution is "over-kill". As soon as I saw the "smallest possible value" I automatically kicked into Calculus mode)
I apologize, but as an AI text-based bot, I am unable to open links or view images directly. However, if you provide me with the information or context from the figure, I can help you with your assignments and explain how to solve them.
1. To find the area of triangle AEF, we need to find the relationship between triangle AEF and the other triangles. In the figure, if triangle AEF is formed by subtracting the areas of triangles CEF, ABE, and ADF from the area of triangle ABC, then we can use the formula:
Area of triangle AEF = Area of triangle ABC - Area of triangle CEF - Area of triangle ABE - Area of triangle ADF
Given the areas of triangles CEF, ABE, and ADF as 3, 4, and 5 respectively, substitute these values into the formula to find the area of triangle AEF.
2. To find the sum of the distances from point P to AB, AB, and BC in an equilateral triangle ABC, we need to use the properties of an equilateral triangle. In an equilateral triangle, all sides are equal and all angles are 60 degrees.
Since the side length of the equilateral triangle is 2 and the area is square root of 3, we can use the formula for the area of an equilateral triangle, A = (sqrt(3)/4) * s^2, where s is the side length.
Given that the side length s = 2, we can substitute this value into the formula to find the area of the equilateral triangle. Then, we can calculate the distances from point P to AB, AB, and BC by finding the perpendicular distances from point P to each side of the triangle.
3. To find the smallest possible value of PE + PC in an accompanying figure where ABCD is a square, BE = 3 cm, EC = 1 cm, and P is a point on the diagonal BD, we can use the properties of similar triangles and the Pythagorean theorem.
In this case, we can consider the triangles BPE and CEP as similar triangles, since they share the same angles. By setting up a proportion between their corresponding sides, we can find the values of PE and PC. Then, by adding the values of PE and PC, we can determine the smallest possible value of PE + PC.
4. To find the length of CE in triangle ABC, where AB = 7, AC = 9, BD = 3, and the area of quadrilateral BCED is 5/7 the area of triangle ABC, we need to use the formula for the area of a triangle and the properties of similar triangles.
By calculating the area of triangle ABC and multiplying it by 5/7, we can find the area of quadrilateral BCED. Using this information and the given lengths of AB, AC, and BD, we can set up a proportion and solve for the length of CE.
5. To find the ratio between the area of a square inscribed in a circle and the area of an equilateral triangle circumscribed about the same circle, we can use the properties of the shape and certain mathematical formulas.
In this case, we can find the area of the square inscribed in the circle by calculating the side length of the square and then squaring it. The side length of the square is equal to the diameter of the circle.
Similarly, we can find the area of the equilateral triangle circumscribed about the same circle by using the formula for the area of an equilateral triangle, which involves the side length of the triangle.
By dividing the area of the square by the area of the equilateral triangle, we can determine the ratio between them.