Suppose that two rectangles are similar, meaning that the ratio of their bases is proportional to the ratio of their heights. If their bases and heights are commensurable, prove that the ratio of their areas is proportional to the ratio of the squares of (or on) their bases. By taking halves, the same result obtains for similar triangles (why?).
What ratio am I trying to prove? And then, any help in proving it? Thanks.
2006-08-31 05:04:45 · 4 answers · asked by Jesse 2 in Science & Mathematics ➔ Mathematics
Let h_1 and b_1 denote the height and width of the base of one rectangle and h_2 and b_2 denote the height and width of the base of the other of the other.
Since they are similar, there is a positive number k so that h_1=k b_1 and h_1=k b_2.
The area of the first rectangle is A_1=h_1 b_1=k (b_1)^2 and the area of the second is A_2=h_2 b_2=k (b_2)^2
Then, A_1/A_2=(.k (b_1)^2)/(k (b_2)^2)=(b_1)^2/(b_2)^2.
The questions does seem oddly phrased, but I'm pretty sure that this is what they are looking for. Good luck! :)
2006-08-31 08:56:32 · answer #1 · answered by Anonymous · 0⤊ 0⤋
The ratio of bases and heights being commensurable means that bse1/base2=height1/height2=a/b where a and b are whole numbers.
Find the area of each one, then calculate the ratio of the areas. Then calculate the areas of the squares which have the base of each (or the height of each) as a side length. The ratio of the areas of these squares is the same as the ratio of the areas of the rectangles. Likewise for the triangles.
2006-08-31 12:17:27 · answer #2 · answered by Benjamin N 4 · 0⤊ 0⤋
You're trying to determine that squares and triangles can all have the same area ratio.
If a square has an area ration of 4:4 then the triangles would have an area ratio of 2:2.
Basically half of the square shape!
I THINK!!!
2006-08-31 12:13:06 · answer #3 · answered by Wilf 2 · 0⤊ 0⤋
You're given two rectangles, which I'll call 1 and 2. You are also told that they have bases b1 and b2 and heights h1 and h2. Because they are similar, you know that b1/b2 = h1/h2. Using simple manipulations of that equation, you can get one expression for b1 in terms of b2. You know that the area of the first rectangle, A1, is equal to b1*h1, and A2 = b2*h2. You are trying to show that A1/A2, the ratio of their areas, is equal to b1^2/b2^2, the ratio of the squares of the bases. If you replace b1 in the A1 formula with the expression in terms of b2, and then take the ratio of A1 and A2, you should get a term that contains only h1 and h2. Go back and use the initial relationship, b1/b2 = h1/h2, to convert this into a term that contains only b1 and b2, and you will have shown the desired result.
2006-08-31 12:25:51 · answer #4 · answered by DavidK93 7 · 0⤊ 0⤋