Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out.
The figure APML is a square.
Given: PL and AM are two perpendicular diameters of a circle with centre O.
Reasoning:
- PL and AM are diameters of the same circle, so they are equal in length
- Both diameters pass through centre O, so they bisect each other at O
- The diameters are perpendicular, intersecting at 90°
Deduction: When diagonals are equal, bisect each other, and intersect at right angles, the quadrilateral formed is a square.
∴ APML is a square.
Explanation
The chapter establishes through Deduction 5 that when diagonals are equal and bisect each other at right angles, the quadrilateral formed is a square. In this case, PL and AM are diameters of the same circle, so they are equal. Being diameters, they both pass through the centre O, so they bisect each other at O. Since they are perpendicular, they intersect at 90°. Therefore, APML is a square.
Solution Steps
Identify that PL and AM are diameters of the same circle, so they are equal in length
Note that both diameters pass through centre O, meaning they bisect each other at O
Recognize that the diameters are perpendicular, intersecting at 90°
Apply the property: when diagonals are equal, bisect each other, and intersect at right angles, the quadrilateral is a square
Conclude that APML is a square