### Clarifications

*Clarification 1*: Instruction includes using compass and straightedge, string, reflective devices, paper folding or dynamic geometric software.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Geometric Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Inscribed Circle
- Circumscribed Circle
- Triangle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students used a relationship between triangles and circles to understand the formula for the area of a circle. In Geometry, students identify and construct two special circles that are associated with a triangle.- Instruction includes the use of manipulatives, tools and geometric software. Allowing students to explore constructions with dynamic software reinforces why the constructions work.
- Instruction includes the student understanding that in a geometric construction, one does not use the markings on a ruler or on a protractor to construct inscribed and circumscribed circles of a triangle. Students should realize that there are limitations on precision that are inherent in the markings on rulers or protractors.
- It is important to build the understanding that formal constructions are valid when the lengths of segments or measures of angles are not known, or have values that do not appear on a ruler or protractor, including irrational values.
- Instruction includes the connection to logical reasoning and visual proofs when verifying that a construction works.
- Instruction includes the connection to constructing angle bisectors and perpendicular
bisectors.
*(MTR.2.1)* - Instruction includes using various methods, like the one described below, to construct an
inscribed circle.
- For example, given triangle
*ABC*, students can construct two of the three angle bisectors to create their point of intersection,*D*. Students should realize that the point*D*is the incenter of the triangle and may predict that point*D*will be the center of the inscribed circle. To prove this prediction, students will need to prove that point*D*is equidistant from each of the three sides. In order to prove this, students can construct the perpendicular segments from point*D*to each of the three sides, and show that all three segments are congruent using triangle congruence criteria and*D*is the intersection of the angle bisectors. Each of these segments will be a radius of the inscribed circle, with the center of the circle at point*D*.

- For example, given triangle

- When constructing an inscribed circle, students should make the connection to
constructing perpendicular bisectors when they need to construct a line through the
incenter of the triangle that is perpendicular to a side of the triangle.
- For example, to construct such a line, students can place the compass at the
incenter, point
*D*, and draw arcs to determine two points,*E*and*F*, on one of the sides. These points are equidistant to*D*. Then they, using the same compass setting, place the compass at*E*and at*F*and draw arcs intersecting on the opposite side of*EF*from*D*. The intersection of these arcs,*Q*, is the same distance to*E*and to*F*. Therefore, the line passing thru*D*and*Q*is the perpendicular bisector of*EF*so it is also a line perpendicular to the side of the triangle.

- For example, to construct such a line, students can place the compass at the
incenter, point
- Students should understand that the shortest segment from a point,
*D*, to a line is the segment from*D*to the line that is perpendicular to the line. Additionally, students should understand that the circle centered at point*D*, which has this segment as a radius, is tangent to the line. - Instruction includes using various methods, like the one described below, to construct a
circumscribed circle.
- For example, given triangle
*ABC*, students can construct two of the three perpendicular bisectors of the sides of the triangle to create their point of intersection,*D*. Students should realize that the point*D*is the circumcenter of the triangle and may predict that point*D*will be the center of the circumscribed circle. To prove this prediction, students will need to prove that point*D*is equidistant from each of the three vertices. In order to prove this, students can use the fact that point*D*is the intersection of the perpendicular bisectors. Each of these segments will be a radius of the circumscribed circle, with the center of the circle at point*D*. So, to construct the circumscribed circle, one can set the compass equal to the distance between point*D*and any one of the vertices and then draw the circle centered at point*D*.

- For example, given triangle

- Instruction includes exploring the construction of circumscribed circles about various triangles. Have students explore acute, right and obtuse triangles, and compare the locations of the circumcenter of each. Students should understand that with a right triangle, the circumcenter is located at the midpoint of the hypotenuse.
- For expectations of this benchmark, constructions should be reasonably accurate and the emphasis is to make connections between the construction steps and the definitions, properties and theorems supporting them.
- While going over the steps of geometric constructions, ensure that students develop
vocabulary to describe the steps precisely.
*(MTR.4.1)* - Problem types include identifying the next step of a construction, a missing step in a construction or the order of the steps in a construction.

### Common Misconceptions or Errors

- Students may think that the when constructing a circumscribed circle, the center of the circle cannot be outside the triangle.

### Instructional Tasks

*Instructional Task 1 (*

*MTR.2.1*,*MTR.4.1*)- Part A. Construct angle bisectors for the three interior angles of a triangle using folding paper, a compass and straightedge and geometric software. What do you notice about each method of construction?
- Part B. Repeat Part A with a triangle that is obtuse, isosceles, acute and right. Describe your findings.
- Part C. Using the incenter as the center, a circle can be constructed inscribed in the triangle. How can you determine the radius of that circle, the inscribed circle?
- Part D. Construct the inscribed circle of one of the triangles from Part B.

Instructional Task 2 (

Instructional Task 2 (

*MTR.2.1*,*MTR.4.1*)- Part A. Construct perpendicular bisector for the three sides of a triangle using folding paper, a compass and straightedge and geometric software. What do you notice about each method of construction?
- Part B. Repeat Part A with a triangle that is obtuse, isosceles, acute and right. Describe your findings.
- Part C. Using the circumcenter as the center, a circle can be constructed circumscribed about the triangle. How can you determine the radius of that circle, the circumscribed circle?
- Part D. Construct the circumscribed circle of one of the triangles from Part B.

Instructional Task 3 (MTR.5.1)

Instructional Task 3 (MTR.5.1)

- Part A. Given the line $l$ and the point
*P*external to the line $l$, construct a perpendicular line, $m$, through point*P*. - Part B. Use the construction from Part A to construct a line, $n$, that is parallel to the line $l$ and
contains the point
*P*.

### Instructional Items

*Instructional Item 1*

- Construct the circle that is circumscribed about Δ
*XYZ*.

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Problem-Solving Tasks

## MFAS Formative Assessments

Students are asked to use a compass and straightedge to construct a circumscribed circle of an acute scalene triangle.

Students are asked to use a compass and straightedge to construct an inscribed circle of an acute scalene triangle.

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.

## Original Student Tutorials Mathematics - Grades 9-12

Learn the steps to circumscribe a circle around a triangle in this interactive tutorial about constructions. Grab a compass, straightedge, pencil and paper to follow along!

Discover how easy it is for Katie to construct an inscribed circular logo on her company's triangular pennant template. If she completes the task first, she will win a $1000 bonus! Follow along with this interactive tutorial.

## Student Resources

## Original Student Tutorials

Discover how easy it is for Katie to construct an inscribed circular logo on her company's triangular pennant template. If she completes the task first, she will win a $1000 bonus! Follow along with this interactive tutorial.

Type: Original Student Tutorial

Learn the steps to circumscribe a circle around a triangle in this interactive tutorial about constructions. Grab a compass, straightedge, pencil and paper to follow along!

Type: Original Student Tutorial

## Problem-Solving Tasks

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 solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).

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 asks students to place a fire hydrant so that it is equal distance from three given points.

Type: Problem-Solving Task

This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).

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