How to determine whether a straight line belongs to a circle. Tangent to a circle
First, let's understand the difference between a circle and a circle. To see this difference, it is enough to consider what both figures are. These are an infinite number of points on the plane, located at an equal distance from a single central point. But, if the circle also consists of internal space, then it does not belong to the circle. It turns out that a circle is both a circle that limits it (circle(r)), and an innumerable number of points that are inside the circle.
For any point L lying on the circle, the equality OL=R applies. (The length of the segment OL is equal to the radius of the circle).
A segment that connects two points on a circle is its chord.
A chord passing directly through the center of a circle is diameter this circle (D). The diameter can be calculated using the formula: D=2R
Circumference calculated by the formula: C=2\pi R
Area of a circle: S=\pi R^(2)
Arc of a circle is called that part of it that is located between its two points. These two points define two arcs of a circle. The chord CD subtends two arcs: CMD and CLD. Identical chords subtend equal arcs.
Central angle An angle that lies between two radii is called.
Arc length can be found using the formula:
- Using degree measure: CD = \frac(\pi R \alpha ^(\circ))(180^(\circ))
- Using radian measure: CD = \alpha R
The diameter, which is perpendicular to the chord, divides the chord and the arcs contracted by it in half.
If the chords AB and CD of the circle intersect at the point N, then the products of the segments of the chords separated by the point N are equal to each other.
AN\cdot NB = CN\cdot ND
Tangent to a circle
Tangent to a circle It is customary to call a straight line that has one common point with a circle.
If a line has two common points, it is called secant.
If you draw the radius to the tangent point, it will be perpendicular to the tangent to the circle.
Let's draw two tangents from this point to our circle. It turns out that the tangent segments will be equal to one another, and the center of the circle will be located on the bisector of the angle with the vertex at this point.
AC = CB
Now let’s draw a tangent and a secant to the circle from our point. We obtain that the square of the length of the tangent segment will be equal to the product of the entire secant segment and its outer part.
AC^(2) = CD \cdot BC
We can conclude: the product of an entire segment of the first secant and its external part is equal to the product of an entire segment of the second secant and its external part.
AC\cdot BC = EC\cdot DC
Angles in a circle
The degree measures of the central angle and the arc on which it rests are equal.
\angle COD = \cup CD = \alpha ^(\circ)
Inscribed angle is an angle whose vertex is on a circle and whose sides contain chords.
You can calculate it by knowing the size of the arc, since it is equal to half of this arc.
\angle AOB = 2 \angle ADB
Based on a diameter, inscribed angle, right angle.
\angle CBD = \angle CED = \angle CAD = 90^ (\circ)
Inscribed angles that subtend the same arc are identical.
Inscribed angles resting on one chord are identical or their sum is equal to 180^ (\circ) .
\angle ADB + \angle AKB = 180^ (\circ)
\angle ADB = \angle AEB = \angle AFB
On the same circle are the vertices of triangles with identical angles and a given base.
An angle with a vertex inside the circle and located between two chords is identical to half the sum of the angular values of the arcs of the circle that are contained within the given and vertical angles.
\angle DMC = \angle ADM + \angle DAM = \frac(1)(2) \left (\cup DmC + \cup AlB \right)
An angle with a vertex outside the circle and located between two secants is identical to half the difference in the angular values of the arcs of the circle that are contained inside the angle.
\angle M = \angle CBD - \angle ACB = \frac(1)(2) \left (\cup DmC - \cup AlB \right)
Inscribed circle
Inscribed circle is a circle tangent to the sides of a polygon.
At the point where the bisectors of the corners of a polygon intersect, its center is located.
A circle may not be inscribed in every polygon.
The area of a polygon with an inscribed circle is found by the formula:
S = pr,
p is the semi-perimeter of the polygon,
r is the radius of the inscribed circle.
It follows that the radius of the inscribed circle is equal to:
r = \frac(S)(p)
The sums of the lengths of opposite sides will be identical if the circle is inscribed in a convex quadrilateral. And vice versa: a circle fits into a convex quadrilateral if the sums of the lengths of opposite sides are identical.
AB + DC = AD + BC
It is possible to inscribe a circle in any of the triangles. Only one single one. At the point where the bisectors of the internal angles of the figure intersect, the center of this inscribed circle will lie.
The radius of the inscribed circle is calculated by the formula:
r = \frac(S)(p) ,
where p = \frac(a + b + c)(2)
Circumcircle
If a circle passes through each vertex of a polygon, then such a circle is usually called described about a polygon.
At the point of intersection of the perpendicular bisectors of the sides of this figure will be the center of the circumcircle.
The radius can be found by calculating it as the radius of the circle that is circumscribed about the triangle defined by any 3 vertices of the polygon.
There is the following condition: a circle can be described around a quadrilateral only if the sum of its opposite angles is equal to 180^( \circ) .
\angle A + \angle C = \angle B + \angle D = 180^ (\circ)
Around any triangle you can describe a circle, and only one. The center of such a circle will be located at the point where the perpendicular bisectors of the sides of the triangle intersect.
The radius of the circumscribed circle can be calculated using the formulas:
R = \frac(a)(2 \sin A) = \frac(b)(2 \sin B) = \frac(c)(2 \sin C)
R = \frac(abc)(4 S)
a, b, c are the lengths of the sides of the triangle,
S is the area of the triangle.
Ptolemy's theorem
Finally, consider Ptolemy's theorem.
Ptolemy's theorem states that the product of diagonals is identical to the sum of the products of opposite sides of a cyclic quadrilateral.
AC \cdot BD = AB \cdot CD + BC \cdot AD
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Mathematics
The figure shows a circle (o, 2) and several segments. Name the radius, chords and diameter of this circle in this figure. Does the circle belong to its center? Does the center belong to the circle? Indicate whether the following statements are true: a) all radii of a given circle are equal b) the radius of a circle is its chord c) the chord of a circle contains exactly two of its points d) the diameter of a circle is its diameter e) the chord of a circle is its diameter. How many radii does a circle have? How many diameters does a circle have? How many diameters can be drawn from a given point on a circle? How many chords can be drawn from a given point on the circle? Is every chord of a circle its diameter?
Answer(s) to the question:
Radii: OC, OD, OA Diameters: CD Chords: AB, CD Does a circle have its center? No Does the center belong to the circle? Yes Indicate whether the following statements are true: a) all radii of a given circle are equal b) a chord of a circle contains exactly two of its points d) the diameter of a circle is its diameter How many radii does a circle have? Unlimited quantity How many diameters does a circle have? Unlimited number How many diameters can be drawn from a given point on a circle? One How many chords can be drawn from a given point on the circle? Unlimited quantity Is every chord of a circle its diameter? No (AB is a chord, but not a diameter)
Dictation 1
1. Complete the sentence.
1) All points of the circle are removed at the same distance from...(from its center ).
2) The radius of a circle is the segment connecting...(its center with a point on the circle).
3) A chord is a segment...(connecting two points on a circle ).
4) Diameter is called...(the largest chord ).
5) The diameter is greater than the radius in...(twice ).
6) An arc of a circle is called each of the parts into which it is divided...(point on a circle).
7) A circle is a part of a plane...(limited by a circle or, as children write, together with a circle).
8) A point belongs to a circle if it is removed from its center at a distance less than...(radius ).
9) A sector is called each of the parts of the circle into which it is divided...(two radii ) .
10) Each of the two parts is called a semicircle after drawing...(diameter ) .
2. Write down what the diameter of the circle is if the distance from the center of the circle to a point belonging to the circle is 8 cm (16 cm ).
3. Does the circle have its center?(No )
4. Does the circle have its center?(Yes )
5. Draw an arbitrary circle. Draw the radius of the circle,
its diameter, on which the drawn radius does not lie, and a chord distinct from the diameter.
6. Inside the circle, mark a point different from its center. How many
through this point you can draw:
1) diameters(one ); 2) chords other than diameter?(infinitely many )
7. Mark an arbitrary point on the circle. How much can you
lead: 1) diameters with an end at this point (one ); 2) chords other than the diameter, with an end at this point (infinitely many ).
Dictation 2
Caliber is the internal diameter of the bore of any weapon. The caliber of the Kalashnikov AK-74 assault rifle is 5.45 mm, and that of the American M-16 assault rifle is 5.56 mm. What percentage is the caliber of the AK-74 smaller than that of an American assault rifle?(≈2% ).
If the caliber of the Msta-S self-propelled gun is 152 mm, then how many centimeters is the diameter of the gun? (15.2 cm ).
How much percent cheaper is the modern Russian T-14 Armata tank from the American Abrams tank, if the Russian one costs 5 million dollars, and the American one costs 10 million? (50% ).
By what percentage is the Abrams heavier than the Armata, if the T-14 weighs 48 tons, and the Abrams weighs 63 tons? (≈31% ).
The caliber of the Kalashnikov AKM assault rifle is 7.62 mm. How much will it be in meters? (0.00762 m ).
If the diameter of a circle is 50.6 cm, then what is its radius? (25,3 cm ).
Draw a segment 6 cm long. Construct a circle so that this segment has a diameter.
Draw a circle of arbitrary radius. Mark three points that lie on the circle and three points that do not lie on it.
Mark an arbitrary point O on the plane. Mark four points 3 cm away from point O. How many more such points can you mark? (infinitely many - they form a circle of radius 3 cm ).
How many axes of symmetry does a circle have? Circle? (infinitely many ).
What is the axis of symmetry for a circle? (any diameter ).
Is it possible to construct a triangle with sides 2 cm, 6 cm and 9 cm? (No ).
“Circle 7th grade” - Constructing the bisector of an angle. Introductory conversation “In the world of circles.” Working with a textbook to study the material. Constructions with compasses and ruler. Any two points on a circle divide it into two parts. The circle has one friend. A segment connecting two points on a circle is called its chord. Circle of arbitrary radius.
“Circle and Circle” - Circle. MATH-5 Thematic planning Lesson progress Author Resources. Favorite activity is reading. Part of a circle is called an arc. Training exercises. The point is called the center of the circle. Arc. Category - highest.
"Circle Length" - Euler. R – radius of the circle. Circle. The great scientist of Ancient Greece Archimedes. Circumference. The more I know, the more I can do. The great mathematician Euler. Ancient Egypt. D is the diameter of the circle. In Ancient Rome they believed that?? 3.12. Archimedes. Ancient Rome. Practical work “Measuring coffee cans”.
“Tangent to a circle” - Point of tangency. Tangent sign. Let us prove that if AK and AM are tangent segments, then AK = AM, ?OAK = ? OAM. A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Proof. Let d be the distance from the center O to the straight line KM. KM – tangent? d = R. Tangent property.
"Circle Equation" - Draw a circle for which CD is the radius. Fill the table. Center coordinates: (;) R = equation of circle: Draw a circle for which CD is the diameter. Let a circle be given. Check whether points A(1;?1), B(0;8), C(?3;?1) lie on the circle defined by the equation (x + 3)2 + (y? 4)2 = 25.
“Circle 8th grade” - Corollaries: Let’s draw perpendiculars OK, OL and OM to sides ABC. Theorem. Let us draw the bisectors of the triangle intersecting at point O. A circle can be inscribed in any triangle. Inscribed circle.
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