Calculating Square Area Based on Diagonal and Area Relations

This problem involves understanding the relationship between the diagonal and area of a square, and how areas scale. We are given the diagonal of square A and the area relationship between square A and square B. We need to find the area of a new square whose side is the diagonal of square B.

Step-by-Step Solution

Understanding the Relationship Between Diagonal and Area of a Square

For any square with side length 's', the diagonal 'd' can be found using the Pythagorean theorem: $d^2 = s^2 + s^2 = 2s^2$. So, $d = s\sqrt{2}$.

The area of the square is $A = s^2$.

We can express the area in terms of the diagonal. From $d = s\sqrt{2}$, we get $s = \frac{d}{\sqrt{2}}$. Substituting this into the area formula:

\( A = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{(\sqrt{2})^2} = \frac{d^2}{2} \)

So, the area of a square is half the square of its diagonal.

Step 1: Find the Area of Square A

The diagonal of square A is given as \((a + b)\) units.

Using the formula $A = \frac{d^2}{2}$, the area of square A is:

\( \text{Area of A} = \frac{(a + b)^2}{2} \)

Step 2: Find the Area of Square B

The problem states that the area of square B is twice the area of square A.

\( \text{Area of B} = 2 \times \text{Area of A} \)

\( \text{Area of B} = 2 \times \frac{(a + b)^2}{2} \)

\( \text{Area of B} = (a + b)^2 \)

Step 3: Find the Diagonal of Square B

Let the side length of square B be \(s_B\). The area of square B is \(s_B^2\).

\( s_B^2 = (a + b)^2 \)

\( s_B = \sqrt{(a + b)^2} = a + b \) (assuming a+b is positive, which is implied as it's a length)

The diagonal of square B, let's call it \(d_B\), is related to its side length by \(d_B = s_B\sqrt{2}\).

\( d_B = (a + b)\sqrt{2} \)

Step 4: Find the Area of the Square Drawn on the Diagonal of Square B

We need to find the area of a new square whose side length is equal to the diagonal of square B.

The side length of this new square is \(s_{new} = d_B = (a + b)\sqrt{2}\).

The area of this new square is \(s_{new}^2\).

\( \text{Area of new square} = ((a + b)\sqrt{2})^2 \)

\( \text{Area of new square} = (a + b)^2 \times (\sqrt{2})^2 \)

\( \text{Area of new square} = (a + b)^2 \times 2 \)

\( \text{Area of new square} = 2(a + b)^2 \)

Thus, the area of the square drawn on the diagonal of square B is \(2(a + b)^2\) square units.

The calculated area matches option 3: \(2(a + b)^2\).

Revision Table: Key Concepts

Additional Information: Properties of Squares

A square is a special type of quadrilateral with four equal sides and four right angles (90 degrees). Key properties include:

• All four sides are equal in length.
• All four interior angles are right angles.
• Opposite sides are parallel.
• Diagonals are equal in length.
• Diagonals bisect each other at right angles.
• Diagonals bisect the angles (each forms two 45-degree angles).
• A square is both a rhombus (all sides equal) and a rectangle (all angles 90 degrees).
• The diagonal of a square divides it into two congruent right-angled isosceles triangles.

Understanding these properties helps in solving geometry problems involving squares and their diagonals.