The side length of square ABCD is 1 unit. Its diagonal, AC, is a side of square ACEF. Square ACEF is then enlarged by a scale factor of 2. What is the area of the enlargement, in square units?
diagonal of first square = √2
so sides of second square are √2
third square has sides 2√2
area of third square = (2√2)^2 = 8 square units.
To find the area of the enlargement, we need to first find the area of square ACEF and then multiply it by the scale factor of 2 squared.
Step 1: Find the area of square ACEF
Since the side length of square ABCD is 1 unit, the diagonal AC of square ABCD is √(1^2 + 1^2) = √2 units (using the Pythagorean theorem).
Step 2: Enlarge the square ACEF by a scale factor of 2
This means that each side of the square ACEF will be multiplied by 2.
The side length of the enlarged square ACEF will be 2 * √2 units.
Step 3: Find the area of the enlarged square ACEF
The area of a square is calculated by squaring its side length.
The area of the enlarged square ACEF = (2 * √2)^2
= 4 * 2
= 8 square units.
Therefore, the area of the enlargement is 8 square units.
To find the area of the enlargement, we need to determine the side length of the enlarged square ACEF.
Since the original square ABCD has a side length of 1 unit, we can find the length of the diagonal AC using the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In our case, AC is the hypotenuse, and the two legs of the right triangle are AB and BC, both with a length of 1 unit.
Using the Pythagorean theorem, we can calculate the length of the diagonal AC:
AC^2 = AB^2 + BC^2
AC^2 = 1^2 + 1^2
AC^2 = 2
Taking the square root of both sides, we find:
AC = √2
So, the length of the diagonal AC is √2.
Now, since square ACEF has AC as one of its sides, the length of the side of the enlarged square ACEF is also √2.
When a square is enlarged by a scale factor of 2, the area is increased by the square of the scale factor. In this case, the scale factor is 2, so the area of the enlargement is (2^2) times the area of the original square ACEF.
The area of the original square ACEF is (side length)^2, which is (√2)^2 = 2.
Therefore, the area of the enlargement is (2^2) * 2 = 4 * 2 = 8 square units.
Hence, the area of the enlargement is 8 square units.