Geometry

Step 1: Recall the Pythagorean theorem for a right-angled triangle.
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 (legs).
If are the lengths of the legs and is the length of the hypotenuse, then .
The hypotenuse is always the longest side.

Step 2: Check each option.
Option (1) Sides: 5 cm, 12 cm, 13 cm.
Longest side is 13 cm (potential hypotenuse).
Check: .
Hypotenuse squared: .
Since (i.
e.
, ), these sides can form a right-angled triangle.
This is a Pythagorean triple (5, 12, 13).
This statement is TRUE.
Option (2) Sides: 4 cm, 4 cm, 10 cm.
Longest side is 10 cm.
Check: .
Hypotenuse squared: .
Since , these sides cannot form a right-angled triangle.
(Also, triangle inequality: , so these can't even form a triangle).
Option (3) Sides: 3 cm, 3 cm, 5 cm.
Longest side is 5 cm.
Check: .
Hypotenuse squared: .
Since , these sides cannot form a right-angled triangle.
Option (4) Sides: 6 cm, 6 cm, 10 cm.
Longest side is 10 cm.
Check: .
Hypotenuse squared: .
Since , these sides cannot form a right-angled triangle.

Step 3: Identify the set of sides that forms a right-angled triangle.
Option (1) satisfies the Pythagorean theorem.
This matches option (1).

Step 1: Define the properties of the geometric shapes involved.
- Parallelogram: A quadrilateral with two pairs of parallel sides.
Opposite sides are equal in length, and opposite angles are equal.
- Rectangle: A parallelogram with all four angles being right angles (90 degrees).
- Square: A rectangle with all four sides equal in length.
(Alternatively, a rhombus with all right angles).
- Rhombus: A parallelogram with all four sides equal in length.
Opposite angles are equal, and diagonals bisect each other at right angles.

Step 2: Evaluate each statement.
Option (1) All rectangles are parallelograms: A rectangle has two pairs of parallel sides (by definition, as it's a special type of parallelogram).
This statement is TRUE.
Option (2) All rectangles are squares: A square is a rectangle where all sides are equal.
A rectangle can have unequal adjacent sides (e.
g.
, length = 5, width = 3).
So, not all rectangles are squares.
This statement is FALSE.
Option (3) All rectangles are Rhombuses: A rhombus has all four sides equal.
A rectangle can have unequal adjacent sides.
So, not all rectangles are rhombuses (unless it's a square, which is both).
This statement is FALSE.
Option (4) All squares are not parallelograms: A square has two pairs of parallel sides (as it's a special rectangle, which is a special parallelogram).
So, all squares are parallelograms.
This statement is FALSE.

Step 3: Identify the true statement.
The only true statement is "All rectangles are parallelograms.
" This matches option (1).