In The Circle, We Are Four...
So multiply this by 4 " 2. Tony B. Apr 25, Answer: Further comment. Explanation: There are a set of 'special' numbers that crop up all over the place and are very useful. One of them is pi pi is the number you get if you divide the circumference of a circle by its diameter. Part of which is: 3.
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The dots at the end mean that the digits go on a lot further. Hope this helps By the way: when you round a decimal it is good practice to state the number of decimal places it is rounded to.
Related questions How do I determine the molecular shape of a molecule? What is the lewis structure for co2? What is the lewis structure for hcn? How is vsepr used to classify molecules? Complete the rectangle ACBR.
Impact of increasing the radius
Because ACBR is a rectangle, its diagonals bisect each other and are equal. A set of points in the plane is often called a locus. The term is used particularly when the set of points is the curve traced out by a moving point. For example, a circle can be defined as the locus of a point that moves so that its distance from some fixed point is constant. The two examples below use the converse of the angle in a semicircle theorem to describe a locus. A photographer is photographing the ornamental front of a building. He wants the two ends of the front to subtend a right angle at his camera.
Describe the set of all positions where he can stand. A plank of length metres is initially resting flush against a wall, but it slips outwards, with its top sliding down the wall and its foot moving at right angles to the wall. What path does the midpoint of the plank trace out? Angles at the centre and circumference. The angle-in-a-semicircle theorem can be generalised considerably. In each diagram below, AB is an arc of a circle with centre O , and P is a point on the opposite arc.
We shall show that this relationship holds also for the other two cases, when the arc is a minor arc left-hand diagram or a major arc right-hand diagram. An angle at the circumference of a circle is half the angle at the centre subtended by the same arc. Let AB be an arc of a circle with centre O, and let P be any point on the opposite arc.
The proof divides into three cases, depending on whether:.
A punter stands on the edge of a circular racing track. With his binoculars he is following a horse that is galloping around the track at one revolution a minute. Hence the punter is rotating his binoculars at a constant rate that is half the rate at which the horse is rotating about the centre. This corollary of the previous theorem is a particularly significant result about angles in circles:.
Some alternative terminology. The last two theorems are often expressed in slightly different language, and some explanation is needed to avoid confusion. An altitude of a triangle is a perpendicular from any of the three vertices to the opposite side, produced if necessary.
The two cases are illustrated in the diagrams below. There are three altitudes in a triangle. The following theorem proves that they concurrent at a point called the orthocentre H of the triangle. It is surprising that circles can be used to prove the concurrence of the altitudes. The altitudes of a triangle are concurrent. In the module, Congruence , we showed how to draw the circumcircle through the vertices of any triangle.
To do this, we showed that the perpendicular bisectors of its three sides are concurrent, and that their intersection, called the circumcentre of the triangle, is equidistant from each vertex. No other circle passes through these three vertices. If we tried to take as centre a point P other than the circumcentre, then P would not lie on one of the perpendicular bisectors, so it could not be equidistant from the three vertices.
When there are four points, we can always draw a circle through any three of them provided they are not collinear , but only in special cases will that circle pass through the fourth point. A cyclic quadrilateral is a quadrilateral whose vertices all lie on a circle. This is the last type of special quadrilateral that we shall consider. Suppose that we are given a quadrilateral that is known to be cyclic, but whose circumcentre is not shown perhaps it has been rubbed out.
The circumcentre of the quadrilateral is the circumcentre of the triangle formed by any three of its vertices, so the construction to the right will find its circumcentre. The distinctive property of a cyclic quadrilateral is that its opposite angles are supplementary. The following proof uses the theorem that an angle at the circumference is half the angle at the centre standing on the same arc.
Join the radii OB and OD. Here is an alternative proof using the fact that two angles in the same segment are equal. An exterior angle of a cyclic quadrilateral is supplementary to the adjacent interior angle, so is equal to the opposite interior angle. This gives us the corollary to the cyclic quadrilateral theorem:. An exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
This exterior angle and A are both supplementary to BCD , so they are equal. Show that AP CR in the diagram to the right. If a cyclic trapezium is not a rectangle, show that the other two sides are not parallel, but have equal length. The property of a cyclic quadrilateral proven earlier, that its opposite angles are supplementary, is also a test for a quadrilateral to be cyclic. That is the converse is true. If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.
Construct the circle through A , B and D , and suppose, by way of contradiction, that the circle does not pass through C. If an exterior angle of a quadrilateral equals the opposite interior angle, then the quadrilateral is cyclic. In the diagram to the right, the two adjacent acute angles of the trapezium are equal.
Prove that the trapezium is cyclic. The sine rule states that for any triangle ABC , the ratio of any side over the sine of its opposite angle is a constant,. Each term is the ratio of a length over a pure number, so their common value seems to be a length. Thus it reasonable to ask, what is this common length? The proof of this result provides a proof of the sine rule that is independent of the proof given in the module, Further Trigonometry.
It is sufficient to prove that is the diameter of the circumcircle. A tangent to a circle is a line that meets the circle at just one point. The diagram below shows that given a line and a circle, can arise three possibilities:. The point where a tangent touches a circle is called a point of contact. It is not immediately obvious how to draw a tangent at a particular point on a circle, or even whether there may be more than one tangent at that point.
Let T be a point on a circle with centre O. First we prove parts a and c. Let be the line through T perpendicular to the radius OT. Let P be any other point on , and join the interval OP. Hence P lies outside the circle, and not on it. This proves that the line is a tangent, because it meets the circle only at T. It also proves that every point on , except for T , lies outside the circle.
It remains to prove part b , that there is no other tangent to the circle at T. Let t be a tangent at T , and suppose, by way of contradiction, that t were not perpendicular to OT. Hence U also lies on the circle, contradicting the fact that t is a tangent. Using this radius and tangent theorem, and the angle in a semi circle theorem, we can now construct tangents to a circle with centre O from a point P outside the circle. It is also a simple consequence of the radius-and-tangent theorem that the two tangents PT and PU have equal length.
The right angle formed by a radius and tangent gives further opportunities for simple trigonometry.
Features of a circle from its standard equation | Analytic geometry (video) | Khan Academy
Show that. The following exercise involves quadrilaterals within which an incircle can be drawn tangent to all four sides. These quadrilaterals form yet another class of special quadrilaterals. The sides of a quadrilateral are tangent to a circle drawn inside it. Show that the sums of opposite sides of the quadrilateral are equal.
A line that is tangent to two circles is called a common tangent to the circles. When the points of contact are distinct, there are two cases, as in the diagrams below. The two circles lie on the same side of a direct common tangent , and lie on opposite sides of an indirect common tangent. Two circles are said to touch at a common point T if there is a common tangent to both circles at the point T. As in the diagram below, the circles touch externally when they are on opposite sides of the common tangent, and touch internally when they are on the same side of the common tangent.
Provided that they are distinct, touching circles have only the one point in common. We are now in a position to prove a wonderful theorem on the angle bisectors of a triangle. These three bisectors are concurrent, and their point of intersection is called the incentre of the triangle. The incentre is the centre of the incircle tangent to all three sides of the triangle, as in the diagram to the right. The angle bisectors of a triangle are concurrent, and the resulting incentre is the centre of the incircle, that is tangent to all three sides.
This completes the development of the four best-known centres of a triangle. The results, and the associated terminology and notation, are summarised here for reference. In the next two diagrams, the angle BQU remains equal to P as the point Q moves around the arc closer and closer to A. In the last diagram, Q coincides with A , and AU is a tangent.
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