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**Question 1**
* **Question Stem:** ABCD is a square. AC=12cm. What is the area of square ABCD in cm²?
* **Options:**
A) 72
B) 80
C) 100
D) 150
* **Chart/Diagram Description:**
* **Type:** Geometric figure (Square)
* **Main Elements:**
* A square labeled with vertices A, B, C, D in counterclockwise order.
* A diagonal line connecting vertex A to vertex C.
* The diagonal AC is labeled with the value "12".
**Question 2**
* **Question Stem:** ABCD is a square. The perimeter of square ABCD is 64 cm. What is the area of square ABCD in cm²?
* **Options:**
A) 121
B) 256
C) 324
D) 440
* **Chart/Diagram Description:**
* **Type:** Geometric figure (Square)
* **Main Elements:**
* A square labeled with vertices A, B, C, D in counterclockwise order.
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We have a square ABCD with diagonal AC equal to 12 centimeters. We need to find the area of this square. Let's start by understanding the relationship between a square's diagonal and its side length.
In any square, there's a special relationship between the diagonal and the side length. If the side length is s, then the diagonal equals s times the square root of 2. This comes from the Pythagorean theorem applied to the right triangle formed by two sides and the diagonal.
Now let's solve for the side length. We know the diagonal AC equals 12 centimeters. Using our formula, 12 equals s times square root of 2. To find s, we divide 12 by square root of 2. Rationalizing the denominator, we get s equals 6 times square root of 2 centimeters.
Now we can calculate the area. The area of a square equals the side length squared. So the area equals 6 square root of 2, all squared. This gives us 36 times 2, which equals 72 square centimeters. Therefore, the answer is A, 72.
For question 2, we use the perimeter to find the area. Given that the perimeter is 64 centimeters, and knowing that perimeter equals 4 times the side length, we get 64 equals 4s. Solving for s, we find the side length is 16 centimeters. The area is then 16 squared, which equals 256 square centimeters. The answer is B, 256.