Question

Prove Proposition III-3, that if a diameter of a circle bisects a chord, then it is perpendicular to the chord. And if a diameter is perpendicular to a chord, then it bisects the chord.

Short answer

Short Answer: If a diameter bisects a chord, it creates two congruent triangles using the SSS congruence postulate (with the radii, bisected chord, and common side being congruent), and their angles must sum to 180°, making the diameter perpendicular to the chord. If a diameter is perpendicular to a chord, it forms two congruent triangles via the hypotenuse-leg congruence, and since corresponding sides are congruent, the chord must be bisected.

Step-by-step solution

Construct the triangles

Let O be the center of the circle, and let AB be a chord that is bisected by the diameter OD. Let E be the midpoint of AB, so that AE = EB. We have two triangles, ΔAEO and ΔBEO.

Prove the triangles are congruent

We know that AO = BO because they are both radii of the circle, and OE = OE (common side). Since we are given that the diameter bisects the chord, we have AE = EB. By the SSS congruence postulate, ΔAEO ≅ ΔBEO.

Show that the angles between the diameter and chord are right angles

Since the triangles are congruent, the angles within the triangles are also congruent. Specifically, we have ∠AEO ≅ ∠BEO. We also know that ∠AEO + ∠BEO = 180° as they form a straight line. Therefore, each angle must be 90°, proving that the diameter is perpendicular to the chord. Proof that if a diameter is perpendicular to a chord, then it bisects the chord:

Construct the triangles

Now we are given that the diameter OD is perpendicular to chord AB at point E. We again have two triangles, ΔAEO and ΔBEO.

Prove the triangles are congruent

We know that AO = BO because they are radii, and OE = OE (common side) is shared by the two triangles. Since ∠AEO and ∠BEO are right angles (given), we have ΔAEO ≅ ΔBEO by the hypotenuse-leg (HL) congruence.

Show that the chord is bisected by the diameter

Since the triangles are congruent, the corresponding sides are congruent. Therefore, AE = EB. Thus, the diameter bisects the chord AB.

Key concepts

SSS Congruence Postulate

The SSS (Side-Side-Side) Congruence Postulate is a fundamental principle in geometry. It states that if three sides of one triangle are equal in length to the corresponding three sides of another triangle, then the two triangles are congruent. Congruence of triangles means that the triangles have the exact shape and size; they are identical in all respects.

In the provided exercise, this postulate is essential for proving the relationship between a diameter that bisects a chord and perpendicularity. By establishing that the segments AE and EB are equal, and knowing that the radii AO and BO are inherently equal in a circle, we can assert that triangles ΔAEO and ΔBEO have three pairs of equal sides. Therefore, by the SSS postulate, they are congruent.

Triangle Congruency

Triangle congruency refers to the condition where two triangles are exact 'copies' of each other in terms of size and shape, even if one is a mirror image or is rotated. Congruent triangles have all corresponding sides and angles equal. There are several ways to prove triangle congruency, and one of the most common is the SSS Congruence Postulate mentioned earlier. Other rules include SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and HL (Hypotenuse-Leg) for right-angled triangles.

In our exercise, once we established the congruence of triangles ΔAEO and ΔBEO through the SSS Postulate, the corresponding parts of congruent triangles (CPCTC) principle allows us to state that all their angles and sides are congruent, which leads us to deduce that the angles ∠AEO and ∠BEO are right angles.

Perpendicular Diameter

A perpendicular diameter in a circle is a line drawn from the center of the circle to the circumference so that it makes a right angle (90 degrees) with a chord of the circle. Establishing that a diameter is perpendicular to a chord involves showing that it satisfies the property of forming right angles with the chord.

In the context of our exercise, by proving that triangles ΔAEO and ΔBEO are congruent and their corresponding angles are right angles, we have demonstrated that the diameter OD is in fact perpendicular to the chord AB. This perpendicular relationship has extensive applications in geometric constructions and proofs, in particular, revealing that certain lines and angles within a circle have special and predictable relationships.

Chord Bisecting

When we say that a chord is bisected, we mean that a line (in this case, a diameter of the circle) cuts the chord into two equal lengths. This concept is central to many geometric theorems involving circles. To prove that a diameter bisects a chord, one typically demonstrates that the two segments of the chord are congruent.

In the exercise, once we've proven that the triangles ΔAEO and ΔBEO are congruent using either the SSS Congruence Postulate or by the HL congruence in the case of right angles, we can infer that their corresponding sides AE and EB are equal. This directly leads to the conclusion that the diameter OD not only is perpendicular to the chord AB but also bisects it into two equal segments, AE and EB. This bisection is a pivotal property that is leveraged in numerous circle theorems and helps us understand how lines and circles interrelate.