Question Video: Finding a Missing Length Using Equidistant Chords from the Center of a Circle | Nagwa Question Video: Finding a Missing Length Using Equidistant Chords from the Center of a Circle | Nagwa

# Question Video: Finding a Missing Length Using Equidistant Chords from the Center of a Circle Mathematics • Third Year of Preparatory School

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Given that ππΆ = ππΉ = 3 cm, π΄πΆ = 4 cm, line segment ππΆ β₯ line segment π΄π΅, and line segment ππΉ β₯ line segment π·πΈ, find the length of the line segment π·πΈ.

03:11

### Video Transcript

Given that ππΆ is equal to ππΉ, which is equal to three centimeters, π΄πΆ is equal to four centimeters, line segment ππΆ is perpendicular to line segment π΄π΅, and line segment ππΉ is perpendicular to line segment π·πΈ, find the length of the line segment π·πΈ.

We are told in the question that ππΆ is equal to ππΉ. And this means that the two chords π΄π΅ and π·πΈ are equidistant from the center π. They are both three centimeters away from the center. We recall the theorem that states that if two chords are equidistant from the center, they are also equal in length. This means that in this question, the length π΄π΅ is equal to the length π·πΈ. We also know that ππΆ is the perpendicular bisector of π΄π΅. And this means that π΄π΅ is equal to two multiplied by π΄πΆ. Since π΄πΆ is equal to four centimeters, π΄π΅ is equal to eight centimeters. We can therefore conclude that the length of the chord π·πΈ is eight centimeters.

So far in this video, we have discussed the lengths of chords depending on their distance from the center of the circle. We will now consider the converse relationship. In the diagram shown, we have two congruent circles. We will consider the case where the chords π΄π΅ and πΆπ· are equal in length and, more importantly, what this says about the distance of the chords from their respective centers. In this example, these are the lengths ππΈ and ππΉ, respectively, Adding the radii ππ΄ and ππΆ, we know that these lengths must be equal, as the circles are congruent. As the chords are equal in length, the half chords must also be equal in length such that π΄πΈ is equal to πΆπΉ. This is because π΄πΈ is equal to a half of π΄π΅ and πΆπΉ is equal to a half of πΆπ·.

Using our knowledge of the Pythagorean theorem, the third sides of our right triangles must also be equal in length. The length ππΈ is equal to the length ππΉ. In other words, the distance of the chords from their respective centers are equal. This can be summarized as follows. Two chords of equal lengths in the same circle or congruent circles are equidistant from the center of the circle or the respective centers.

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