Ptolemy’s Theorem, determine the area of a cyclic quadrilateral as a function of its side lengths and the acute angle formed by its diagonals, prove Ptolemy’s theorem, examples and step by step solutions, Common Core Geometry

Cyclic Quadrilaterals, a quadrilateral is cyclic if and only if its opposite angles are supplementary, derive and apply the area of cyclic quadrilateral ABCD as 1/2 AB·CD·sin(w), examples and step by step solutions, Common Core Geometry

Equations for Tangent Lines to Circles, Given a circle, students find the equations of two lines tangent to the circle with specified slopes, find the equation of the line tangent to the circle from that point, examples and step by step solutions, Common Core Geometry

Recognizing Equations of Circles, complete the square in order to write the equation of a circle in center-radius form, recognize when a quadratic in x and y is the equation for a circle, examples and step by step solutions, Common Core Geometry

Writing the Equation for a Circle, write the equation for a circle in center-radius form using the Pythagorean theorem or the distance formula, examples and step by step solutions, Common Core Geometry

CIE May 2021 9709 Prob & Stats 1 Paper 52, Past Papers, 9709/52, Mark Scheme, step by step solutions

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Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams, find missing lengths in circle-secant or circle-secant-tangent diagrams, examples and step by step solutions, Common Core Geometry

Secant Angle Theorem, Exterior Case, find the measures of angle/arcs and chords in figures that include two secant lines meeting outside a circle, examples and step by step solutions, Common Core Geometry

Secant Lines; Secant Lines That Meet Inside a Circle, an angle whose vertex lies in the interior of a circle intersects the circle in two points and that the edges of the angles are contained within two secant lines of the circle, angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle, examples and step by step solutions, Common Core Geometry

The Inscribed Angle Alternate—A Tangent Angle, use the inscribed angle theorem to prove other theorems in its family, solve missing angle problems using the inscribed angle theorem, solve a variety of missing angle problems using the inscribed angle theorem, examples and step by step solutions, Common Core Geometry

Tangent Segments, use tangent segments and radii of circles to conjecture and prove geometric statements, use the fact if a circle is tangent to both rays of an angle, then its center lies on the angle bisector, examples and step by step solutions, Common Core Geometry

Properties of Tangents, a line is tangent to a circle at a given point if it is perpendicular to the radius drawn to that point, construct tangents to a circle through a given point, examples and step by step solutions, Common Core Geometry

Unknown Length and Area Problems, apply their understanding of arc length and area of sectors to solve problems of unknown area and length, examples and step by step solutions, Common Core Geometry

Arc Length and Areas of Sectors, how to determine the length of an arc and the area of a sector, examples and step by step solutions, Common Core Geometry

Arcs and Chords, Congruent chords have congruent arcs, Arcs between parallel chords are congruent, examples and step by step solutions, Common Core Geometry

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The Angle Measure of an Arc, Define the angle measure of arcs, and understand that arcs of equal angle measure are similar, The measure of an inscribed angle is half the angle measure of its intercepted arc, Explain and understand that all circles are similar, examples and step by step solutions, Common Core Geometry

Unknown Angle Problems with Inscribed Angles in Circles, Use the inscribed angle theorem to find the measures of unknown angles, examples and step by step solutions, Common Core Geometry

Inscribed Angle Theorem and Its Applications, Recognize and use different cases of the inscribed angle theorem embedded in diagrams, examples and step by step solutions, Common Core Geometry

Experiments with Inscribed Angles, relationship between inscribed angles and central angles and their intercepted arcs, examples and step by step solutions, Common Core Geometry

Rectangles Inscribed in Circles, symmetries of inscribed rectangles across a diameter, examples and step by step solutions, Common Core Geometry

Circles, Chords, Diameters, and Their Relationships, Identify the relationships between the diameters of a circle and other chords of the circle, examples and step by step solutions, Common Core Geometry

Thales’ Theorem, explore the converse of Thales' theorem, Prove the statement of Thales' theorem, examples and step by step solutions, Common Core Geometry

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The Distance from a Point to a Line, name several points on a line given by a parametric equation and provide the point-slope equation for a line given by a parametric equation, determine whether lines given parametrically are parallel or perpendicular, examples and step by step solutions, Common Core Geometry

Motion Along a Line—Search Robots Again, name several points on a line given by a parametric equation and provide the point-slope equation for a line given by a parametric equation, determine whether lines given parametrically are parallel or perpendicular, examples and step by step solutions, Common Core Geometry

2021 Physics 9702 Paper 22, Past Papers, 9702/22, Mark Scheme, step by step solutions

October 2020 Physics 9702 Paper 23, Past Papers, 9702/23, Mark Scheme, step by step solutions

Analytic Proofs of Theorems Previously Proved by Synthetic Means, prove that the intersection of the medians of a triangle meet at a point that is two-thirds of the way along each median from the intersected vertex, prove the diagonals of a parallelogram bisect one another and meet at the intersection of the segments joining the midpoints of opposite sides, examples and step by step solutions, Common Core Geometry

CIE May 2021 9709 Prob & Stats 2 Paper 63, Past Papers, 9709/63, Mark Scheme, step by step solutions

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Dividing Segments Proportionately, find midpoints of segments and points that divide segments into 3, 4, or more proportional, equal parts, examples and step by step solutions, Common Core Geometry

Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities, find the perimeter of a triangle or quadrilateral in the coordinate plane given a description by inequalities, find the area of a triangle or quadrilateral in the coordinate plane given a description by inequalities by employing Green’s theorem, examples and step by step solutions, Common Core Geometry

Perimeter and Area of Polygonal Regions in the Cartesian Plane, find the perimeter of a quadrilateral in the coordinate plane given its vertices and edges, find the area of a quadrilateral in the coordinate plane given its vertices and edges by employing Green’s theorem, examples and step by step solutions, Common Core Geometry

Perimeter and Area of Triangles in the Cartesian Plane, find the perimeter of a triangle in the coordinate plane using the distance formula, state and apply the formula for area of a triangle with vertices, examples and step by step solutions, Common Core Geometry

Parallel and Perpendicular Lines, recognize parallel and perpendicular lines from slope, examples and step by step solutions, Common Core Geometry

Equations for Lines Using Normal Segments, apply the criterion to determine if two segments are perpendicular, examples and step by step solutions, Common Core Geometry

Segments That Meet at Right Angles, apply the criterion to determine if two segments are perpendicular, examples and step by step solutions, Common Core Geometry

Criterion for Perpendicularity, explain the connection between the Pythagorean Theorem and the criterion for perpendicularity, examples and step by step solutions, Common Core Geometry

Designing a Search Robot to Find a Beacon, find the segments obtained by rotating the given segment by 90° counterclockwise and clockwise about one endpoint, examples and step by step solutions, Common Core Geometry

Lines That Pass Through Regions, Given two points in the coordinate plane and a rectangular or triangular region, students determine whether the line through those points meets the region, examples and step by step solutions, Common Core Geometry

Finding Systems of Inequalities That Describe Triangular and Rectangular Regions, describe rectangles (with edges parallel to the axes) and triangles in the coordinate plane by means of inequalities, examples and step by step solutions, Common Core Geometry

Searching a Region in the Plane, Visualize cross-sections of three-dimensional objects, how a 3D printer works and its relation to Cavalieri’s principle, examples and step by step solutions, Common Core Geometry

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