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Clarifications
Clarification 1: Instruction includes cases in which a triangle inscribed in a circle has a side that is the diameter.Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Circle
- Circumscribed Circle
- Diameter
- Inscribed Polygon in a Circle
- Quadrilateral
- Triangle
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 8, students learned about regular polygons and their interior angles. In Algebra 1, students rearranged formulas to highlight a quantity of interest and solved linear equations in one variable. In Geometry, students learn about the properties of the interior angles of triangles and quadrilaterals that are inscribed in a circle and how they relate to one another.- Instruction includes using precise definitions and language when working with triangles and quadrilaterals inscribed in a circle. Students should be able to determine the similarities and differences between each and how their relationships interact with one another. (MTR.4.1)
- For example, students should be able to answer questions like “What is the difference between a polygon inscribed in a circle and a polygon circumscribed about a circle?” and “What is the difference between a circle inscribed in a polygon and a polygon inscribed in a circle?”
- Instruction includes the connection to constructing equilateral triangles and squares inscribed in a circle.
- Instruction includes exploring various triangles inscribed in a circle.
- For example, when one side of a triangle inscribed in a circle is the diameter of the circle, then the triangle is inscribed in a semicircle. Students should determine what kind of angles are the interior angles of the triangle inscribed in a semicircle and should be able to identify them as inscribed angles. Students can make the connection to the angle opposite the diameter divides the circle into two equal parts, therefore the inscribed angle is one-half of 180°. This conclusion can lead students to Thales’ Theorem which states that the angle inscribed in semicircle is always a right angle. That is, if A, B and C are distinct points on a circle where AC is a diameter, the angle ABC is a right angle. Therefore, the triangle ABC is always a right triangle.
- Instruction includes the connection to the relationships between inscribed angles and their intercepted arcs, when a triangle is inscribed in a circle, and the Triangle Sum Theorem. When a triangle is inscribed in a circle, the three arcs intercepted by the interior angles complete a circle, so the sum of the measures of the interior angles of the triangle is half the measure of a circle in degrees, 180 °. (MTR.5.1)
- A quadrilateral inscribed in a circle is also called a cyclic quadrilateral. The opposite angles of such quadrilateral are supplementary, and each exterior angle is equal to the interior opposite angle. To prove the first of these properties, students can determine what type of angles are the interior angles of the quadrilateral inscribed in a circle. Then students can discover that opposite angles of the quadrilateral have intercepted arcs such that they complete a circle. Using the relationship between inscribed angles and their intercepted arcs, students should be able to conclude that the opposite angles of cyclic quadrilaterals are supplementary.
- Instruction includes the understanding that there are two definitions of an inscribed circle in a polygon. One definition states that the inscribed circle is the largest circle that fits inside the polygon. Under this definition, all polygons have inscribed circles, with some polygons (e.g., rectangles) having more than one inscribed circle. The other definition states that the inscribed circle is the circle that fits inside of the polygon and touches all of its sides. Under this definition, only special polygons (e.g., regular polygons) have inscribed circle. Students can explore the differences by constructing an inscribed circle in a quadrilateral.
- For example, given a quadrilateral that is not a rectangle, students can extend two opposite sides to form a triangle and construct the inscribed circle of that triangle. Students can repeat this for the other two opposite sides of the quadrilateral. Students should be able to reason that one of the two constructed circles is the largest circle that fits inside the given quadrilateral.
- Extension of this benchmark includes the introduction of Ptolemy’s Theorem with cyclic quadrilaterals. This theorem states that the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides. Students should realize that when this theorem is applied to a rectangle inscribed in a circle, it can be used to prove the Pythagorean Theorem.
- For example, given rectangle ABCD inscribed in circle O. Students can apply Ptolemy’s Theorem, so that AC⋅ BD = AB ⋅ CD + AD ⋅ BC. Since the opposite sides of a rectangle are congruent and the diagonals of a rectangle are congruent, then AB = CD, AD = BC and AC = BD. Using the Substitution Property of Equality, then AC ⋅ AC = AB ⋅ AB + BC ⋅ BC. That is AC2 = AB2 + BC2.
- For example, given rectangle ABCD inscribed in circle O. Students can apply Ptolemy’s Theorem, so that AC⋅ BD = AB ⋅ CD + AD ⋅ BC. Since the opposite sides of a rectangle are congruent and the diagonals of a rectangle are congruent, then AB = CD, AD = BC and AC = BD. Using the Substitution Property of Equality, then AC ⋅ AC = AB ⋅ AB + BC ⋅ BC. That is AC2 = AB2 + BC2.
Common Misconceptions or Errors
- Students may think that every quadrilateral can be inscribed in a circle. To help address this misconception, have students use a triangle inscribed in a circle to see that there are many places a fourth vertex can be placed to form a quadrilateral that cannot be inscribed in the circle.
Instructional Tasks
Instructional Task 1 (MTR.3.1)- Quadrilateral DCFE is inscribed in Circle A and the measure of angle D is 75°.
- Part A. What is the measure of angle CAE? What is the measure of angle CFE?
- Part B. What is the measure of arc CDE?
- Part C. What can you determine about the measures of angle DCF and angle FED?
Instructional Items
Instructional Item 1- In circle A, segment DE is a diameter.
- Part A. Determine the measure of angle C.
- Part B. If the measure of arc CF is 50°, determine the measures of angle D and angle F.
Instructional Item 2
- Triangle DAE is inscribed in Circle K.
- Part A. Determine the value of if the measure of angle E is (2 + 30)°.
- Part B. Determine the measure of angle D if the measure of angle A is (2 − 20)°.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Problem-Solving Tasks
MFAS Formative Assessments
Students are asked to prove that opposite angles of a quadrilateral, inscribed in a circle, are supplementary.
Students are given a grid with three points (vertices of a right triangle) representing the starting locations of three sprinters in a race and are asked to determine the center of the finish circle, which is equidistant from each sprinter.
Student Resources
Problem-Solving Tasks
This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.
Type: Problem-Solving Task
This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.
Type: Problem-Solving Task
This problem solving task asks students to explain certain characteristics about a triangle.
Type: Problem-Solving Task
This problem solving task asks students to place a fire hydrant so that it is equal distance from three given points.
Type: Problem-Solving Task
This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle.
Type: Problem-Solving Task
This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.
Type: Problem-Solving Task
This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Tasks
This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.
Type: Problem-Solving Task
This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.
Type: Problem-Solving Task
This problem solving task asks students to explain certain characteristics about a triangle.
Type: Problem-Solving Task
This problem solving task asks students to place a fire hydrant so that it is equal distance from three given points.
Type: Problem-Solving Task
This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle.
Type: Problem-Solving Task
This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.
Type: Problem-Solving Task
This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.
Type: Problem-Solving Task
