WEBVTT
00:00:01.440 --> 00:00:07.820
π΄π΅ and π΅πΆ are two tangents to the circle π where π΅ and πΆ lie on the circumference.
00:00:08.260 --> 00:00:13.560
ππ΄ is equal to 14 centimeters and the radius of the circle is seven centimeters.
00:00:13.920 --> 00:00:21.200
Find the area of the part between the two tangents and the smaller arc π΅πΆ giving the answer to the nearest square centimeter.
00:00:21.890 --> 00:00:23.690
This might look a little tricky at first.
00:00:23.850 --> 00:00:27.300
But we can break the problem down into several smaller parts.
00:00:27.750 --> 00:00:35.680
First, we see that if we add the radii π΅π and ππΆ onto the diagram, we can form two right-angled triangles.
00:00:35.910 --> 00:00:38.670
Thatβs π΄π΅π and π΄πΆπ.
00:00:39.510 --> 00:00:43.970
This is because the angle between a tangent and a radius is 90 degrees.
00:00:44.160 --> 00:00:49.100
So angles π΄π΅π and π΄πΆπ must both be 90 degrees.
00:00:49.790 --> 00:00:57.160
Once we spot this, we can use right angle trigonometry to work out the measure of angle π΄ππ΅ and π΄ππΆ.
00:00:58.020 --> 00:01:04.210
Drawing triangle π΄ππ΅ out, we were told that ππ΄ is 14 centimeters.
00:01:04.490 --> 00:01:07.840
So the hypotenuse of our triangle is 14 centimeters.
00:01:07.970 --> 00:01:11.990
Itβs the side opposite the right angle and itβs always the longest side in the triangle.
00:01:12.780 --> 00:01:15.540
The radius ππ΅ is seven centimeters.
00:01:15.730 --> 00:01:19.890
Thatβs the adjacent side and itβs the side next to the included angle.
00:01:20.400 --> 00:01:24.460
We can now use the cosine ratio to find the measure of the angle π.
00:01:25.060 --> 00:01:28.380
Cos of π is equal to adjacent divided by hypotenuse.
00:01:28.380 --> 00:01:33.600
In this case, thatβs seven divided by 14 which is 0.5.
00:01:34.230 --> 00:01:39.230
We can solve this equation for π by finding the inverse of the cosine ratio.
00:01:39.230 --> 00:01:42.190
Thatβs inverse cos of 0.5.
00:01:42.740 --> 00:01:46.100
This is one of those standard results that we should know by heart.
00:01:46.300 --> 00:01:49.820
But if we donβt, we get π over three radians.
00:01:50.660 --> 00:01:55.330
Now that we have this information, we can find the area of the triangle π΄π΅π.
00:01:55.850 --> 00:02:01.070
Once we have that, we can then find the area of the quadrilateral π΄π΅ππΆ.
00:02:01.310 --> 00:02:06.170
And then, we will be able to subtract the area of the arc to find the area required.
00:02:06.800 --> 00:02:09.820
There are actually two ways we could find the area of this triangle.
00:02:09.950 --> 00:02:15.390
One way would be to find the length of the missing side and then use the formula a half base times height.
00:02:15.800 --> 00:02:20.260
However, we can use the trigonometric formula: a half π’π£ sin π€.
00:02:20.790 --> 00:02:24.800
The included angle in our triangle π΄π΅π is π over three.
00:02:24.940 --> 00:02:29.300
And the sides π’ and π£ are seven centimeters and 14 centimeters.
00:02:29.730 --> 00:02:35.760
So the area is a half multiplied by seven multiplied by 14 multiplied by sine of π over three.
00:02:36.640 --> 00:02:40.930
Once again, sine of π over three is one of those standard results we should know by heart.
00:02:40.960 --> 00:02:42.200
Its root three over two.
00:02:42.790 --> 00:02:46.110
And we can simplify somewhat by dividing through by two.
00:02:46.560 --> 00:02:54.600
Once we do, we can see that the area of our right-angled triangle is 49 root three over two units squared or square units.
00:02:55.200 --> 00:02:59.310
We can find the area of the quadrilateral then by multiplying this value by two.
00:02:59.560 --> 00:03:02.770
Thatβs two multiplied by 49 root three over two.
00:03:03.370 --> 00:03:08.800
So the area of π΄π΅ππ is 49 root three units squared.
00:03:09.260 --> 00:03:15.390
Remember to find the shaded area, we said weβd need to subtract the area of sector πππ.
00:03:16.140 --> 00:03:22.360
The formula for area of a sector with radius π³ and angle π radians is a half π³ squared π.
00:03:22.970 --> 00:03:25.680
We know the size of the angle in our sector.
00:03:25.680 --> 00:03:28.740
Itβs two lots of π over three, two π over three.
00:03:29.010 --> 00:03:34.170
So the area is a half multiplied by seven squared multiplied by two π over three.
00:03:34.610 --> 00:03:36.250
Seven squared is 49.
00:03:36.280 --> 00:03:38.770
And we can simplify by dividing through by two.
00:03:39.600 --> 00:03:42.960
And the area of our sector is 49 π over three.
00:03:43.680 --> 00:03:46.480
The area we need is the difference between these.
00:03:46.600 --> 00:03:50.640
Itβs 49 root three minus 49 π over three.
00:03:51.100 --> 00:03:53.810
Thatβs 33.557.
00:03:54.010 --> 00:03:57.540
Remember we were asked to give our answer to the nearest square centimeter.
00:03:57.980 --> 00:04:05.170
And if we do, we can see that the area we require is 34 units squared or 34 centimeters squared.