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.

Question

Homi · Accepted Answer

We will look at the properties of the diagonals of the figure APML to identify what kind of shape it is.

Step 1 — Understanding the circle and its diameters

Let us consider a circle with its center at point O. The radius is the distance from the center to any point on the circle. Let the radius of this circle be $r$ . A diameter is a line segment that passes through the center and connects two points on the circle. Its length is always twice the radius. We are told that PL and AM are two diameters of this circle. This means points P, L, A, and M are all on the circle. Since PL and AM are diameters, they both pass through the center O. We are also told that these two diameters, PL and AM, are perpendicular to each other. This means they cross each other at a 90-degree angle at the center O.

Diagram 1

Step 2 — Examining the diagonals of APML

The figure APML is a quadrilateral, which is a shape with four sides. Its vertices are A, P, M, and L. The line segments connecting opposite vertices are called diagonals. For APML, the diagonals are PL and AM. Let us find the lengths of these diagonals. Since PL is a diameter, its length is twice the radius $r$ .

$PL = \text{radius} + \text{radius}$

$PL = r + r$

$PL = 2r$

Similarly, AM is also a diameter, so its length is also twice the radius $r$ .

$AM = \text{radius} + \text{radius}$

$AM = r + r$

$AM = 2r$

So, we can see that the lengths of the diagonals are equal.

$\boxed{PL = AM}$

Next, let us check if the diagonals bisect each other. To bisect means to cut into two equal halves. Since O is the center of the circle, and PL is a diameter, O is the midpoint of PL. This means the distance from O to P is the same as from O to L.

$OP = OL = r$

Similarly, O is the midpoint of AM.

$OA = OM = r$

Since O is the midpoint for both diameters, the diagonals PL and AM bisect each other at point O. Finally, the problem states that PL and AM are perpendicular. This means they cross at a 90-degree angle.

$\boxed{\text{Diagonals are perpendicular}}$

Step 3 — Identifying the quadrilateral APML

We have found three important properties about the diagonals of the quadrilateral APML:

The diagonals are equal in length ( $PL = AM$ ).
The diagonals bisect each other (at point O).
The diagonals are perpendicular to each other.

Let us recall the properties of different quadrilaterals:

A parallelogram has diagonals that bisect each other.
A rectangle has diagonals that bisect each other and are equal in length.
A rhombus has diagonals that bisect each other and are perpendicular.
A square has diagonals that bisect each other, are equal in length, and are perpendicular.

Since APML has all three properties (equal, bisecting, and perpendicular diagonals), it fits the definition of a square perfectly.

Answer

(i) The diagonals PL and AM are equal in length. (ii) The diagonals PL and AM bisect each other at the center O. (iii) The diagonals PL and AM are perpendicular to each other. (iv) Therefore, the figure APML is a square.

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