If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Given that EG is a diameter of ⊙ L and it is perpendicular to DF, we will show that DC≅ FC and DG≅FG.
To do so we will use congruent triangles. Let's consider △ LCD and △ LCF. Since EG ⊥ DF, by the definition of a right angle ∠ LCD and ∠ LCF are right angles.
Therefore, by the definition of a right triangle △ LCD and △ LCF are right triangles. Next, let's remember that the radii of a circle are congruent. Since both LD and LF are radii of ⊙ L, they are also congruent.
LD≅ LF
We can also see that △ LCD and △ LCF share the same leg LC. By the Reflexive Property of Congruence LC is congruent to itself.
LC≅ LC
Combining all of this information, we can see that the hypotenuse and one leg of △ LCD are congruent to the hypotenuse and the corresponding leg of △ LCF.
Thus, by the Hypotenuse Leg (HL) Theorem △ LCD is congruent to △ LCF.
△ LCD ≅ △ LCF
Since Corresponding Parts of Congruent Triangles are Congruent (CPCTC), we can prove the first part of the statement.
DC ≅ FC
Next, again by CPCTC, we can tell that ∠ DLC is congruent to ∠ FLC.
∠ DLC ≅ ∠ FLC
At this point, to prove the second part of the statement we will recall the Congruent Central Angles Theorem (Theorem 10.4).
Congruent Central Angles Theorem
In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.
Finally, by this theorem we can conclude that DG is congruent to FG.
DG ≅ FG
Let's summarize the above process in a flow proof.
Completed Proof
2
&Given:&& EG is a diameter of ⊙ L.
& &&EG⊥ DF
&Prove:&& DC≅ FC, DG≅FG
Proof:
