SOLUTION: 1. A rhombus with diagonals of lengths 18 and 30. What are the area??? 2. An isosceles triangle with sides of lengths 10, 10 and 12. what are the area?? 3.A rectangle has wid

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Question 254769: 1. A rhombus with diagonals of lengths 18 and 30. What are the area???
2. An isosceles triangle with sides of lengths 10, 10 and 12. what are the area??
3.A rectangle has width represented by x and length 4x. if the area of the rectangle is 80. find the value of x?
Answer by dabanfield(803) (Show Source):
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1. A rhombus with diagonals of lengths 18 and 30. What are the area???
2. An isosceles triangle with sides of lengths 10, 10 and 12. what are the area??
3.A rectangle has width represented by x and length 4x. if the area of the rectangle is 80. find the value of x?
1) There is a theorem that says the area of a rhombus is equal to half the product of its diagonals. In this case area = (18*30)/2.
2) If the triangle is ABC and AB = AC = 10 and AC = 12 then we can drop a perpenduclar from the vertex at B bisecting the side AC at point D and also the angle B ( We then have two congruent triangles ABD and CBD. BD = DC = 5. By the Pythagorean Teorem we know then that 10^2 = 5^2 + BD^2.
So BD^2 = 75 = 5*sqrt(3). Since BD is the altitude of the triangle the area is (1/2)*12*(5*sqrt(3)) = 30*sqrt(3).
3) Since the area is length times width we have:
x*(4x) = 80
4*x^2 = 80
x^2 = 20
x = sqrt(20) = sqrt(4*5) = 2*sqrt(5)