WEBVTT
00:00:01.860 --> 00:00:07.410
The line segment π΄π΅ is a chord in circle π whose radius is 25.5 centimeters.
00:00:07.910 --> 00:00:13.770
If the length of π΄π΅ is 40.8 centimeters, what is the length of the line segment π·πΈ?
00:00:15.500 --> 00:00:18.590
The line segment π·πΈ is the segment highlighted in orange.
00:00:18.680 --> 00:00:24.670
Itβs the line that joins the point π·, which is inside the circle, to the point πΈ on the circle circumference.
00:00:25.990 --> 00:00:34.560
Notice that this is a segment of the line ππΈ, which connects the center of the circle to the point on the circumference, and is therefore a radius of the circle.
00:00:35.690 --> 00:00:42.360
The length of the line segment ππΈ then is 25.5 centimeters as this is the radius of the circle given in the question.
00:00:43.930 --> 00:00:50.300
We could therefore work out the length of the line segment π·πΈ by subtracting the length of ππ· from ππΈ.
00:00:50.510 --> 00:00:53.980
So we need to consider how we could work out the length of ππ·.
00:00:55.300 --> 00:00:58.590
Letβs join in a line connecting the points π and π΅ together.
00:00:58.890 --> 00:01:00.460
π is the center of the circle.
00:01:00.580 --> 00:01:02.520
And π΅ is a point on the circumference.
00:01:02.520 --> 00:01:04.690
Itβs the endpoint of our chord π΄π΅.
00:01:04.840 --> 00:01:13.140
And therefore, ππ΅ is the radius of the circle, which means its length is 25.5 centimeters as given in the question.
00:01:14.560 --> 00:01:18.010
We now have a right-angle triangle, triangle ππ·π΅.
00:01:18.090 --> 00:01:20.100
And we know the length of the hypotenuse.
00:01:21.450 --> 00:01:26.650
Weβre also told in the question that the length of the chord π΄π΅ is 40.8 centimeters.
00:01:26.810 --> 00:01:31.010
And we can use this information to work out the length of the line segment π΅π·.
00:01:32.490 --> 00:01:36.670
We know that the perpendicular bisector of a chord passes through the center of the circle.
00:01:37.310 --> 00:01:41.360
Now, the line ππ· or ππΈ does pass through the center of the circle.
00:01:41.360 --> 00:01:42.920
It passes through the point π.
00:01:43.170 --> 00:01:47.130
And from the diagram, we know that it meets the chord π΄π΅ at right angles.
00:01:47.710 --> 00:01:53.240
Therefore, by the converse of this statement, it must be the perpendicular bisector of the chord π΄π΅.
00:01:54.580 --> 00:02:00.170
So this tells us that the lengths of π·π΅ and π·π΄ are equal because the chord has been bisected.
00:02:01.540 --> 00:02:05.540
We can therefore find each of these lengths by halving the length of the chord π΄π΅.
00:02:05.720 --> 00:02:08.640
40.8 divided by two is 20.4.
00:02:09.960 --> 00:02:12.560
We now know two of the lengths in our right-angle triangle.
00:02:12.710 --> 00:02:19.530
So we can apply the Pythagorean theorem in order to work out the third length ππ·, which is one of the two shorter sides.
00:02:20.810 --> 00:02:28.540
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.
00:02:28.830 --> 00:02:35.990
In our triangle, that means that ππ· squared plus 20.4 squared is equal to 25.5 squared.
00:02:36.130 --> 00:02:39.100
And we can solve this equation to find the length of ππ·.
00:02:40.470 --> 00:02:43.860
20.4 squared is 416.16.
00:02:44.000 --> 00:02:47.630
And 25.5 squared is 650.25.
00:02:49.110 --> 00:02:57.560
Subtracting 416.16 from each side of this equation gives that ππ· squared is equal to 234.09.
00:02:58.950 --> 00:03:06.600
To find the value of ππ· then, we need to take the square root of each side of this equation, taking only the positive square root as ππ· is a length.
00:03:06.920 --> 00:03:13.130
We have that ππ· is equal to the square root of 234.09, which is 15.3.
00:03:14.640 --> 00:03:26.340
So now that weβve calculated the length of ππ·, we can return to our calculation for the length of π·πΈ, which we said we were going to work out by subtracting the length of ππ· from ππΈ, which was the radius of the circle.
00:03:27.700 --> 00:03:40.440
Substituting the lengths for ππΈ, 25.5, and ππ·, 15.3, we have that the length of π·πΈ is equal to 25.5 minus 15.3, which gives 10.2.
00:03:40.810 --> 00:03:46.110
The units for this are centimeters because the radius of the circle was also given in centimeters.
00:03:47.520 --> 00:03:58.980
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 π·πΈ is 10.2 centimeters.