Download G.9 - DPS ARE

Geometry — 8.G
ELG.MA.8.G.1 Understand congruence and
similarity using physical models, transparencies, or
geometry software.
 8.G.A.5 Use informal arguments to establish
facts about the angle sum and exterior angle of
triangles, about the angles created when
parallel lines are cut by a transversal, and the
angle-angle criterion for similarity of triangles.
For example, arrange three copies of the same
triangle so that the sum of the three angles
appears to form a line, and give an argument in
terms of transversals why this is so.
Circles – G-C
ELG.MA.HS.G.9: Understand and apply theorems
about circles.
 G-C.A.4 (+) Construct a tangent line from a point
outside a given circle to the circle.
Geometry: Circles — G-C
ELG.MA.HS.G.9: Understand and apply theorems
about circles.
 G-C.A.1 Prove that all circles are similar.
 G-C.A.2 Identify and describe relationships
among inscribed angles, radii, and chords.
Include the relationship between central,
inscribed, and circumscribed angles; inscribed
angles on a diameter are right angles; the
radius of a circle is perpendicular to the tangent
where the radius intersects the circle.
 G-C.A.3 Construct the inscribed and
circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral
inscribed in a circle.
Students will demonstrate command of the ELG by:
 Constructing a tangent line from a point outside a given circle to the circle.
Vocabulary:



circle
construct
tangent line
Sample Assessment Questions:
Standard(s):
G-C.A.4 Source: https://www.illustrativemathematics.org/content-standards/HSG/C/A/4/tasks/1096
Item Prompt:
Suppose C is a circle with center O and P is a point outside of C. Let M be the midpoint of ̅̅̅̅
𝑂𝑃 and let D be the circle with center M passing through O.
Let A and B be the two points of intersection of C and D, pictured below along with several line segments of interest:
a.
Show that angles OAP and OBP are right angles.
b.
Show that ⃡𝑃𝐴 and ⃡𝑃𝐵 are tangent lines from P to the circle C.
Correct Answer(s):
Below is a picture of the different points and triangles used in the solution of the problem:
a.
Segment ̅̅̅̅
𝑂𝑃 is a diameter of circle D since the center of D is the midpoint M of this segment. The points A and B are also both on D since they are the points of
intersection of C and D. The angles OAP and OBP are both right angles because segment ̅̅̅̅
𝑂𝑃 is a diameter of circle D, and A and B are points on D: this means that
triangles OAP and OBP are inscribed in circle D and so the angle opposite the diameter must be a right angle.
b.
Since angles OAP and OBP are right angles it follows that ⃡𝑃𝐴 meets the radius ̅̅̅̅
𝑂𝐴 in a right angle and similarly ⃡𝑃𝐵 meets radius ̅̅̅̅
𝑂𝐵 in a right angle. This means that
⃡ is tangent to C at A and ⃡𝑃𝐵 is tangent to C at B.
𝑃𝐴
Below is a picture with the two tangent lines constructed above: