Understanding the Relationship Between the Rectangle and the Square

Oct 13, 2025 by miodeya.com 112 views

In the following picture a rectangle is in a square. The length of the square side is 20cm. The four rectangular vertices are on the square side so that the length is equal to three times the width. Find broad

In this problem, we are given a square with a side length of 20cm. Inside the square, a rectangle is drawn such that its four vertices touch the sides of the square. The length of the rectangle is three times its width. Our goal is to find the width of the rectangle.

Understanding the Relationship Between the Rectangle and the Square

To solve this problem, we need to understand the relationship between the rectangle and the square. Since the four vertices of the rectangle touch the sides of the square, we can draw a diagonal line from one corner of the rectangle to the opposite corner. This diagonal line will also be a diagonal of the square.

Drawing a Diagram

Let's draw a diagram to visualize the problem. We can draw a square with a side length of 20cm and a rectangle inside it. The length of the rectangle is three times its width. We can label the width of the rectangle as 'x' and the length as '3x'.

Finding the Width of the Rectangle

Since the diagonal of the rectangle is also a diagonal of the square, we can use the Pythagorean theorem to find the width of the rectangle. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Applying the Pythagorean Theorem

In this case, the diagonal of the rectangle is the hypotenuse of a right-angled triangle formed by the width and length of the rectangle. We can write the Pythagorean theorem as:

x^2 + (3x)^2 = 20^2

Simplifying the Equation

We can simplify the equation by expanding the left-hand side:

x^2 + 9x^2 = 400

Combining Like Terms

We can combine like terms by adding the two terms on the left-hand side:

10x^2 = 400

Dividing Both Sides by 10

We can divide both sides of the equation by 10 to isolate x^2:

x^2 = 40

Taking the Square Root of Both Sides

We can take the square root of both sides of the equation to find x:

x = √40

Simplifying the Square Root

We can simplify the square root by factoring 40:

x = √(4 × 10)

Simplifying Further

We can simplify further by taking the square root of 4:

x = √10

Finding the Width of the Rectangle

The width of the rectangle is x, which is equal to √10.

In this problem, we were given a square with a side length of 20cm and a rectangle inside it. The length of the rectangle was three times its width. We used the Pythagorean theorem to find the width of the rectangle, which is equal to √10.

Q: What is the relationship between the rectangle and the square?

A: The four vertices of the rectangle touch the sides of the square, forming a diagonal line that is also a diagonal of the square.

Q: How can we use the Pythagorean theorem to find the width of the rectangle?

A: We can use the Pythagorean theorem to find the width of the rectangle by considering the right-angled triangle formed by the width and length of the rectangle. The diagonal of the rectangle is the hypotenuse of this triangle.

Q: What is the equation we get after applying the Pythagorean theorem?

A: The equation we get after applying the Pythagorean theorem is x^2 + (3x)^2 = 20^2.

Q: How can we simplify the equation?

A: We can simplify the equation by expanding the left-hand side and combining like terms.

Q: What is the final equation we get after simplifying?

A: The final equation we get after simplifying is 10x^2 = 400.

Q: How can we find the value of x?

A: We can find the value of x by dividing both sides of the equation by 10 and then taking the square root of both sides.

Q: What is the final answer for the width of the rectangle?

A: The final answer for the width of the rectangle is √10.

Q: Can we simplify the square root of 10?

A: Yes, we can simplify the square root of 10 by factoring 10 as 4 × 10 and then taking the square root of 4.

Q: What is the simplified form of the square root of 10?

A: The simplified form of the square root of 10 is √(4 × 10) = √4 × √10 = 2√10.

Q: What is the approximate value of the square root of 10?

A: The approximate value of the square root of 10 is 3.162.

Q: Can we use a calculator to find the value of the square root of 10?

A: Yes, we can use a calculator to find the value of the square root of 10.

Q: What is the decimal form of the square root of 10?

A: The decimal form of the square root of 10 is 3.16227766.

Q: Can we use the square root of 10 to find the length of the rectangle?

A: Yes, we can use the square root of 10 to find the length of the rectangle by multiplying the width by 3.

Q: What is the length of the rectangle?

A: The length of the rectangle is 3√10 or approximately 9.486.

Q: Can we use the Pythagorean theorem to find the area of the rectangle?

A: Yes, we can use the Pythagorean theorem to find the area of the rectangle by squaring the length and width and then multiplying them together.

Q: What is the area of the rectangle?

A: The area of the rectangle is (3√10) × (√10) = 30.

Q: Can we use the Pythagorean theorem to find the perimeter of the rectangle?

A: Yes, we can use the Pythagorean theorem to find the perimeter of the rectangle by adding the length and width and then multiplying by 2.

Q: What is the perimeter of the rectangle?

A: The perimeter of the rectangle is 2(3√10 + √10) = 20√10 or approximately 62.82.