The line segment 𝐴𝐵 is a chord in
circle 𝑀 whose radius is 25.5 centimeters. If the length of 𝐴𝐵 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
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 𝐵 together. 𝑀 is the center of the circle. And 𝐵 is a point on the
circumference. It’s the endpoint of our chord
𝐴𝐵. And therefore, 𝑀𝐵 is the radius
of the circle, which means its length is 25.5 centimeters as given in the
We now have a right-angle triangle,
triangle 𝑀𝐷𝐵. And we know the length of the
hypotenuse. We’re also told in the question
that the length of the chord 𝐴𝐵 is 40.8 centimeters. And we can use this information to
work out the length of the line segment 𝐵𝐷.
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 𝐴𝐵 at right angles. Therefore, by the converse of this
statement, it must be the perpendicular bisector of the chord 𝐴𝐵. So this tells us that the lengths
of 𝐷𝐵 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 𝐴𝐵. 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.