Consider a circle with centre . Line segments and are two perpendicular diameters of the circle. What is the figure ? Reason and/or experiment to figure this out.

Question

Homi · Accepted Answer

Understand the Question

The line segments $PL$ and $AM$ are two perpendicular diameters of a circle with centre $O$ and radius $r$ .
Joining the four endpoints on the circle forms the quadrilateral $APML$ , where the diameters $PL$ and $AM$ serve as its diagonals.
We can determine the type of quadrilateral by analyzing three properties of its diagonals: their lengths, whether they bisect each other, and the angle between them.

Step 1 · Analyze the Diagonals of APML

Let the radius of the circle be $r$ . Diagram 1

Since $PL$ and $AM$ are diameters of the circle:

\begin{aligned} PL &= \text{radius} + \text{radius} \\ &= r + r \\ &= 2r \end{aligned}

\begin{aligned} AM &= \text{radius} + \text{radius} \\ &= r + r \\ &= 2r \end{aligned}

Therefore, the diagonals are equal: $PL = AM$

Since $O$ is the centre of the circle, it is the midpoint of both diameters: $OP = OL = r$ $OA = OM = r$

Thus, the diagonals $PL$ and $AM$ bisect each other at $O$ .

Given that the diameters are perpendicular: $PL \perp AM$

Step 2 · Identify the Shape of APML

For the quadrilateral $APML$ , the diagonals satisfy three conditions:

They are equal in length ( $PL = AM = 2r$ ).
They bisect each other at point $O$ .
They intersect at right angles ( $PL \perp AM$ ).

A quadrilateral whose diagonals are equal, bisect each other, and are perpendicular is a square.

Therefore, the figure $APML$ is a square.

Answer

The figure $APML$ is a square.

Common Mistakes

Confusing Rhombus with Square: A rhombus has perpendicular bisecting diagonals, but its diagonals are not necessarily equal. Since $PL = AM$ , the figure is specifically a square.
Confusing Rectangle with Square: A rectangle has equal bisecting diagonals, but they are not perpendicular unless the rectangle is a square.

Question 3

More questions in FIO