### Clarifications

*Clarification 1:*Problems include relationships between two chords; two secants; a secant and a tangent; and the length of the tangent from a point to a circle.

**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

- Circle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In middle grades, students learned about circles, including the definition of center, radius (radii), diameter and circumference, and they used the Pythagorean Theorem to find lengths of segments. In Algebra 1, students rearranged formulas to highlight a quantity of interest and solved linear and quadratic equations in one variable. In Geometry, students learn about the lengths and relationships between a variety of line segments involving circles.- Instruction includes using precise definitions and language when working with segments involving circles. Students should be able to determine the similarities and differences between each of the various segments 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 chord and diameter?”, “Is a diameter always a chord?” and “What is the difference between tangent and secant?”

- Instruction includes the understanding that while tangents and secants are often defined
as lines, when determining lengths of these, one is referring to just a segment of the tangent or secant. The endpoints of these segments are typically a point of intersection between two lines, a point of intersection between a line and the circle, or a given point that is external to the circle.
*(**MTR.4.1*) - Instruction includes the connection to similarity criteria and the Inscribed Angle Theorem to prove, and understand, the relationship between two chords. Typically, the theorem that describes the relationship between two lengths of chords is called the Intersecting Chords Theorem or the Chord-Chord Theorem.
- For example, given the circle shown below with chords
*AD*and*CB*, two triangles are formed, Δ*APB*and Δ*CPD*. Students should notice that angle*APB*and angle*CPD*are congruent because they are vertical angles. Students should also notice that angle ABC and angle ADC are congruent because they intercept the same arc on the circle. Therefore, using the Angle-Angle criterion, students can prove that

- For example, given the circle shown below with chords

- these two triangles are similar. Since corresponding sides are proportional, students can determine that
**$\frac{\text{AP}}{\text{CP}}$**Using this fact, students can rearrange the formula to state that*= $\frac{\text{BP}}{\text{DP}}$*.*AP ⋅ DP = CP ⋅ BP,*which is the Intersecting Chords Theorem. - Instruction includes the connection to similarity criteria and the Inscribed Angle Theorem
to prove, and understand, the relationship between two secants. The Intersecting Secants Theorem, or the Secant-Secant Theorem, states that the product of the length of an entire secant segment and its external segment is equal to the product of the length of another entire secant and its external segment.
- For example, given the circle below, the Intersecting Secants Theorem states that
*PA ⋅ PB*=*PC ⋅ PD*. This can be proved using Angle-Angle criterion to prove that Δ*ADP~*Δ*CBP.*

- For example, given the circle below, the Intersecting Secants Theorem states that

- Instruction includes relating the Intersecting Secants Theorem to the Tangent-Secant Theorem and the Tangent-Tangent Theorem. Students can build the understanding that the Tangent-Secant Theorem and the Tangent-Tangent Theorem are specific cases of the Intersecting Secants Theorem.
- For example, to prove the Tangent-Secant Theorem, using the above circle,
students can move points
*C*and*D*toward one another until they meet at a tangent point,*T*, creating the tangent segment*TP*. Students then can determine that*PA ⋅ PB*=*PT ⋅ PT*which is equivalent to*PA ⋅ PB*=*PT*.^{2} - For example, to prove the Tangent-Tangent Theorem, using the above circle, students can move points C and D toward one another until they meet at a tangent point,
*T*, creating the tangent segment*TP*. Likewise, students can move points*A*and*B*towards one another until they meet at a tangent point,*SP*, creating the tangent segment*TP*. Students then can determine that*PA ⋅ PB*=*PT ⋅ PT*which is equivalent to*PA . PB*=*PT*.^{2}

- For example, to prove the Tangent-Secant Theorem, using the above circle,
students can move points
- Instruction includes the connection to properties of perpendicular bisectors to prove, and understand, the relationship between a chord and the diameter of the circle that is perpendicular to the chord.
- Instruction includes the connection to the Pythagorean Theorem to prove, and understand, the relationship between a tangent segment, the radius of the circle to the point of tangency and the line segment from the center of the circle to the external point.
- For example, given the circle below, students can use the Pythagorean Theorem to show that $t$
^{2 }+ $r$^{2}= $l$^{2}. Students can then rearrange the formula to highlight the length of the tangent segment in terms of the length of the radius and the distance between the center and the external point.

- For example, given the circle below, students can use the Pythagorean Theorem to show that $t$

### Common Misconceptions or Errors

- Students may confuse the exterior portion of a secant and the whole secant when determining lengths or products of lengths.

### Instructional Tasks

*Instructional Task 1 (*

*MTR.4.1*, MTR.5.1)- Circle
*C*is shown below with various lines and line segments.*PS*and*PT*are tangent to Circle*C*.

- Part A. Write a statement of equality that involves the length of a tangent and the length of a secant.
- Part B. Write a statement of equality that involves the lengths of two tangents.
- Part C. Write a statement of equality based on the relationship between a chord and a diameter.
- Part D. Write a statement of equality that involves the length of a tangent, the length of a line from the center to the external point and the length of a radius.
- Part E. Compare your statements from Parts
*A, B, C*and*D*with a partner.

Instructional Task 2 (MTR.3.1)

Instructional Task 2 (MTR.3.1)

- In Circle
*A, AE = DE, FE*= 6 inches and*GE*= 10 inches. What is the length of the radius of Circle*A*?

### Instructional Items

*Instructional Item 1*

- In Circle
*A, DE and BC*intersect at point*F. FE*= 1.3 units,*BF*= 1.9 units,*FD = $x$*+ 1.3 units and*CF = $x$*units. Find the value of $x$.

**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 Assessment

## Lesson Plans

## Problem-Solving Tasks

## MFAS Formative Assessments

Students are asked to draw a circle, a tangent to the circle, and a radius to the point of tangency. Students are then asked to describe the relationship between the radius and the tangent line.

## Student Resources

## Problem-Solving Tasks

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Type: Problem-Solving Task

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Type: Problem-Solving Task

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Type: Problem-Solving Task

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Type: Problem-Solving Task

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Type: Problem-Solving Task

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Type: Problem-Solving Task

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Type: Problem-Solving Task

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Type: Problem-Solving Task

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Type: Problem-Solving Task