Squares - Math

Let x represent the length of the original square in inches. Thus the area of the original square is x2. Two inches are added to x, which is represented by x+2. The area of the resulting square is (x+2)2. We are given that the new square is 64 sq. inches greater than the original. Therefore we can write the algebraic expression:

x2 + 64 = (x+2)2

FOIL the right side of the equation.

x2 + 64 = x2 + 4x + 4

Subtract x2 from both sides and then continue with the alegbra.

64 = 4x + 4

64 = 4(x + 1)

16 = x + 1

15 = x

Therefore, the length of the original square is 15 inches.

If you plug in the answer choices, you would need to add 2 inches to the value of the answer choice and then take the difference of two squares. The choice with 15 would be correct because 172 -152 = 64.

The area of a square is equal to length times width or length squared (since length and width are equal on a square). Therefore, the length of one side is l = $$\sqrt{25in^{2}$$} or l=5 in. The perimeter of a square is the sum of the length of all 4 sides or 4 times 5 in. =20 in.

Given that the square's diagonals pass through the circle's center, those diagonals must each form a diameter of the circle. The circle's diameter is twice its radius, i.e. , which is . Since this diameter (i.e., the square's diagonal) is the hypotenuse of a right triangle formed by two sides of the square, the length of one of the square's sides can be calculated with the Pythagorean Theorem. replace and because the sides of the square must be equal in length. Since the objective is to solve for the square's area, solve for since one side squared will be the square's area.

units squared

Perimeter = side * 4

48 = side * 4

Side = 12

We can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.

Therefore, we can use the Pythagorean Theorem to solve for the diagonal:

A square is also a rhombus, so its area can be calculated as one half the product of its diagonals:

,

where is the common diagonal length.

Since , .

The distance between opposite corners is about 49 meters.

The diagonal line cuts the square into two equal triangles. Their hypotenuse is the diagonal of the square, so we can solve for the hypotenuse.

We need to use the Pythagorean Theorem: , where a and b are the legs and c is the hypotenuse.

The two legs have lengths of 8. Plug this in and solve for c:

Diagonal forms a triangle with adjacent sides . Since this is a square we know this is a right triangle and we can use the Pythagorean Theorem to determine the length of . Sides of length form each of the legs and is the hypotenuse. So the equation looks like this:

Solve for

We can simplify this to

The path from home plate to first base is a side of a perfect square; the path from home plate to second base is a diagonal. As two sides and a diagonal form a triangle, the diagonal measures as long as a side.

The distance to second base from home is times the distance to first base:

You can do this problem in two different ways that lead to the final answer:

1. Pythagorean Theorem

2. Special Triangles (45-45-90)

1. For the first idea, use the Pythagorean Theorem: , where a and b are the side lengths of the square and c is the length of the diagonal.

2. If you know that ALL squares can be made into two special right triangles such that their angles are 45-45-90, then there's a formula you could use:

Let's say that your side length of the square is "a". Then the diagonal of the square (or the hypotenuse of the right triangle) will be .

So using this with a=4:

From the perimeter, we can find the length of each side of the square. The side lengths of a square are equal by definition therefore, the perimeter can be rewritten as,

Then we use the Pythagorean Theorme to find the diagonal, which is the hypotenuse of a right triangle with each leg being a side of the square.

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

For the square given in the question,

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

For the square given in the question,

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

For the square given in the question,

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

For the square given in the question,

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

For the square given in the question,

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

For the square given in the question,

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

For the square given in the question,

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

For the square given in the question,

The diagonal of a square is also a hypotenuse of a right triangle with the side lengths as legs of the triangle.

Use the Pythagorean Theorem to find the length of the diagonal.

For the square given in the question,