We can cut a square diagonally in half to get two triangles. Suppose the
perimeter of the square is 3cm longer than the perimeter of a resulting triangle.
What is the perimeter of the square?
Let a-the side of the square, then
sqrt(a^2+a^2)=a*sqrt(2)-diagonal.
On the condition of the problem
4a=2a+a*sqrt(2)+3
a(2-sqrt(2))=3
a=3/(2-sqrt(2))=3(2+sqrt(2))/2
The perimeter of the square=4a=6(2+sqrt(2))=20.49cm
To solve this problem, we need to set up an equation based on the information given.
Let's say the side length of the square is denoted as "s".
The perimeter of a square is calculated by multiplying the side length by 4, so the perimeter of the square is 4s.
When we cut the square diagonally in half, we obtain two congruent right triangles. In a right triangle, the hypotenuse (the longest side) can be found using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In our case, the two legs of the right triangle are equal to the side length "s" of the square. Therefore, the hypotenuse of each triangle is √(s^2 + s^2) = √2s^2 = s√2.
The perimeter of each triangle is the sum of its three sides, which is s + s + s√2 = 2s + s√2.
Given that the perimeter of the square is 3 cm longer than the perimeter of each triangle, we can set up the equation:
4s = 2s + s√2 + 3.
Let's simplify this equation to find the value of "s".
First, we move all terms containing "s" to one side:
4s - 2s = s√2 + 3.
Combining like terms, we have:
2s = s√2 + 3.
Next, we isolate the square root term:
2s - s√2 = 3.
Factoring out the common factor of "s", we get:
s(2 - √2) = 3.
Finally, we solve for "s" by dividing both sides by (2 - √2):
s = 3 / (2 - √2).
Using a calculator to evaluate this expression, we find that s ≈ 5.84 cm.
Therefore, the perimeter of the square is 4s ≈ 4 * 5.84 ≈ 23.36 cm.