Question Video: Finding the Length of a Chord and the Radius | Nagwa Question Video: Finding the Length of a Chord and the Radius | Nagwa

# Question Video: Finding the Length of a Chord and the Radius Mathematics • Third Year of Preparatory School

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The figure shows a circle with center π. Given that πΈπ΅ = 42, πΈπΆ = 21, and πΈπ΄ = 42, find the length of segment πΈπ· and then the radius of the circle.

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### Video Transcript

The figure shows a circle with center π. Given that πΈπ΅ equals 42, πΈπΆ equals 21, and πΈπ΄ equals 42, find the length of segment πΈπ· and then the radius of the circle.

From the information given, we can deduce that chord π΅π΄ is bisected, because its two parts πΈπ΅ and πΈπ΄ both equal 42. It is therefore relevant to recall the first part of the chord bisector theorem. This theorem tells us that if a straight line passing through the center of a circle bisects a chord in the same circle, the line must be perpendicular to the chord. Therefore, the segment πΆπ·, which passes through the center of our circle, is actually the perpendicular bisector of chord π΅π΄.

Because chords π΅π΄ and πΆπ· are perpendicular, we know that they form four right angles, one of which is the angle π΄πΈπ. To answer this question, we must find the length of segment πΈπ· and the radius. We note that segment ππ· is a radius of the circle.

The only other piece of information we have not considered yet is the length of segment πΈπΆ. We are told that πΈπΆ equals 21. We see from the diagram that segment ππΆ is also a radius of the circle. And the length of ππΆ equals the sum of ππΈ plus πΈπΆ. Since we donβt know the length ππΈ, we will let ππΈ equal π₯. Then, we can say that ππΆ equals π₯ plus 21. Since we know that all radii in a circle have equal length, the length of any radius of our circle can be represented by the eπ₯pression π₯ plus 21. Therefore, radius ππ· equals π₯ plus 21.

We also recognize segment ππ΄ as a radius. So ππ΄ equals π₯ plus 21. At this point, we have an eπ₯pression for each side of a right triangle, specifically triangle π΄πΈπ, which we have highlighted in green. The length of side ππΈ equals π₯, πΈπ΄ equals 42, and ππ΄ equals π₯ plus 21.

From here, we can use the Pythagorean theorem, which states that the sum of the squares of the legs of a right triangle equal the square of the hypotenuse. Substituting the eπ₯pressions for each side length, we have π₯ squared plus 42 squared equals π₯ plus 21 squared. We will proceed to solve this equation for π₯. Once we know the value of π₯, we will be able to find the length of segment πΈπ· and the radius. We find that 42 squared is 1764 and π₯ plus 21 squared is π₯ squared plus 42π₯ plus 441.

Now it is necessary to simplify our quadratic equation into standard forms that it can be solved. However, after subtracting π₯ squared from each side of the equation, we realize that this is no longer a quadratic equation. What remains is the equation 1764 equals 42π₯ plus 441. Solving this equation, we find that π₯ equals 31.5.

Now we are ready to find the exact length of the radius, which we previously found to be represented by the expression π₯ plus 21. So we substitute 31.5 for π₯. And we find that the radius equals 52.5. Finally, to find the length of segment πΈπ·, we will use the fact that πΈπ· equals ππΈ plus ππ·, which we see is true from the diagram. ππΈ equals π₯. And because ππ· is a radius, it equals 52.5. After substituting 31.5 for π₯ and adding 52.5, we find that πΈπ· equals 84.

In conclusion, the length of segment πΈπ· is 84, and the radius equals 52.5.

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