Video: Finding the Perpendicular Distance between a Point on a Circle and Another on a Chord

The line segment 𝐴𝐡 is a chord in circle 𝑀 whose radius is 25.5 cm. If 𝐴𝐡 = 40.8 cm, what is the length of the line segment 𝐷𝐸?

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Video Transcript

The line segment 𝐴𝐡 is a chord in circle 𝑀 whose radius is 25.5 centimeters. If the length of 𝐴𝐡 is 40.8 centimeters, what is the length of the line segment 𝐷𝐸?

The line segment 𝐷𝐸 is the segment highlighted in orange. It’s the line that joins the point 𝐷, which is inside the circle, to the point 𝐸 on the circle circumference. Notice that this is a segment of the line 𝑀𝐸, which connects the center of the circle to the point on the circumference, and is therefore a radius of the circle. The length of the line segment 𝑀𝐸 then is 25.5 centimeters as this is the radius of the circle given in the question.

We could therefore work out the length of the line segment 𝐷𝐸 by subtracting the length of 𝑀𝐷 from 𝑀𝐸. So we need to consider how we could work out the length of 𝑀𝐷. Let’s join in a line connecting the points 𝑀 and 𝐡 together. 𝑀 is the center of the circle. And 𝐡 is a point on the circumference. It’s the endpoint of our chord 𝐴𝐡. And therefore, 𝑀𝐡 is the radius of the circle, which means its length is 25.5 centimeters as given in the question.

We now have a right-angle triangle, triangle 𝑀𝐷𝐡. And we know the length of the hypotenuse. We’re also told in the question that the length of the chord 𝐴𝐡 is 40.8 centimeters. And we can use this information to work out the length of the line segment 𝐡𝐷.

We know that the perpendicular bisector of a chord passes through the center of the circle. Now, the line 𝑀𝐷 or 𝑀𝐸 does pass through the center of the circle. It passes through the point 𝑀. And from the diagram, we know that it meets the chord 𝐴𝐡 at right angles. Therefore, by the converse of this statement, it must be the perpendicular bisector of the chord 𝐴𝐡. So this tells us that the lengths of 𝐷𝐡 and 𝐷𝐴 are equal because the chord has been bisected. We can therefore find each of these lengths by halving the length of the chord 𝐴𝐡. 40.8 divided by two is 20.4.

We now know two of the lengths in our right-angle triangle. So we can apply the Pythagorean theorem in order to work out the third length 𝑀𝐷, which is one of the two shorter sides. The Pythagorean theorem tells us that, in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In our triangle, that means that 𝑀𝐷 squared plus 20.4 squared is equal to 25.5 squared. And we can solve this equation to find the length of 𝑀𝐷. 20.4 squared is 416.16. And 25.5 squared is 650.25. Subtracting 416.16 from each side of this equation gives that 𝑀𝐷 squared is equal to 234.09.

To find the value of 𝑀𝐷 then, we need to take the square root of each side of this equation, taking only the positive square root as 𝑀𝐷 is a length. We have that 𝑀𝐷 is equal to the square root of 234.09, which is 15.3.

So now that we’ve calculated the length of 𝑀𝐷, we can return to our calculation for the length of 𝐷𝐸, which we said we were going to work out by subtracting the length of 𝑀𝐷 from 𝑀𝐸, which was the radius of the circle. Substituting the lengths for 𝑀𝐸, 25.5, and 𝑀𝐷, 15.3, we have that the length of 𝐷𝐸 is equal to 25.5 minus 15.3, which gives 10.2. The units for this are centimeters because the radius of the circle was also given in centimeters.

So by recalling that the perpendicular bisector of a chord passes through the center of the circle and then applying the Pythagorean theorem, we found that the length of the line segment 𝐷𝐸 is 10.2 centimeters.

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