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Picture a square ABCD with sides measuring 20 meters. Now, let’s find the size of the smallest square that fits snugly inside ABCD, with its corners touching the sides of the bigger square.
ABCD is a square of 20 m. What is the area of the least-sized square that can be inscribed in it with its vertices on the sides of ABCD?
The area of the least-sized square that can be inscribed in it with its vertices on the sides of ABCD is 200m².
Diagonal of the larger square:
- Using the Pythagorean theorem:
- Diagonal (d) = √(side^2 + side^2)
- Substituting side length (s) = 20 m:
- d = √(20^2 + 20^2)
- d = √(400 + 400)
- d = √(800)
- d = 20√2 m (approximately 28.28 m)
Side length of the inscribed square:
- Half the diagonal:
- a = d / 2
- a = 20√2 m / 2
- a = 10√2 m (approximately 14.14 m)
Area of the inscribed square:
- Area = side^2
- Area = (10√2 m)^2
- Area = 100 * 2 (since √2^2 = 2)
- Area = 200 m²
Therefore, the area of the least-sized square that can be inscribed in the original square with its vertices on the sides of ABCD is 200 m².
Properties of Square
Here are some of the properties of a square:
- Four equal sides: Each side of a square has the same length. This is the most defining property of a square and what distinguishes it from other quadrilaterals.
- Four right angles: Each corner of a square is a right angle, measuring 90 degrees. This makes a square a special type of rectangle, as all rectangles have right angles, but not all rectangles have equal sides.
- Opposite sides parallel: The opposite sides of a square are parallel to each other. This means that if you extend two opposite sides of a square, they will never intersect.
- Equal diagonals: The diagonals of a square are both equal in length and bisect each other at right angles. This means that they cut each other in the middle at a 90-degree angle.
- Area and perimeter: The area of a square is calculated by squaring the length of one side. The perimeter is the sum of the lengths of all four sides, which is simply 4 times the side length.
- Symmetry: A square has four lines of symmetry, which means that it can be folded or flipped in half in four different ways and still look the same.
Squares are found in many places in the real world, from the tiles on a floor to the windows in a building. They are also used in many mathematical concepts, such as geometry and trigonometry.
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