Let ABCD be a square, and EFGH be a square whose area is twice that of ABCD. Prove that the ratio of AB to AC is equal to the ratio of AB to EF
if the second square is twice the area, then the sides for the large square must be sqrt2 times as large.
EF=AB*sqrt2
EF/AB =sqrt2
Now consider the smaller square. ABC forms a right triangle, sides AB, CD, and AC
but CD=AB so AC by Pythagorean theorem is AC=sqrt(AB^2+AB^2)=sqrt(2AB^2
=AB*sqrt2
AC/AB = sqrt2
so the Ration of AB to AC is the same as ratio of AB to AC
let the side of square ABCD be x
then area = x^2
area of square EFGH = 2x^2
side of square EFGH = √2 x
EF = √2 x
AC^2 = x^2 + x^2 = 2x^2
AC = √2 x
AB/AC = x/√2x = 1/√2
AB/EF = x/√2x = 1/√2
thus AB/AC = AB/EF
To prove the given statement, we need to show that the ratio of the side lengths AB to AC of square ABCD is equal to the ratio of the side lengths AB to EF of square EFGH.
Let's start by determining the side lengths of the squares in terms of a variable. Let x represent the side length of square ABCD and y represent the side length of square EFGH.
The area of a square is equal to the square of its side length. Therefore, the area of square ABCD is x^2, and the area of square EFGH is y^2.
Given that the area of square EFGH is twice that of square ABCD, we have the equation y^2 = 2 * x^2.
Now, let's examine the ratios AB to AC and AB to EF:
AB to AC = x to x = 1 to 1 (since square ABCD has equal side lengths)
AB to EF = x to y
To prove that these ratios are equal, we need to show that x/y = 1.
From the equation y^2 = 2 * x^2, we can rearrange it to obtain y = sqrt(2) * x.
Now, let's substitute this value of y into the ratio AB to EF:
AB to EF = x to (sqrt(2) * x) = 1 to sqrt(2)
Since sqrt(2) is a constant value, we can simplify this ratio as follows:
AB to EF = (1/sqrt(2)) to 1
Now, simplify the ratio:
AB to EF = (1/sqrt(2)) * (sqrt(2)/sqrt(2)) to 1 * (sqrt(2)/sqrt(2)) = 1/sqrt(2) to sqrt(2)/2.
To rationalize the denominator of the left-hand side of the ratio, we multiply the numerator and denominator by sqrt(2):
AB to EF = (1/sqrt(2)) * (sqrt(2)/sqrt(2)) to (sqrt(2)/2) * (sqrt(2)/sqrt(2)) = sqrt(2)/2 to 1.
Comparing this ratio to the AB to AC ratio (which is 1 to 1), we see that both ratios are equal.
Hence, we have proven that the ratio of AB to AC is equal to the ratio of AB to EF in the given scenario.