Tangents and Special Angles

Name Computer Number

Construct Circle A. Label child z.

Choose the circle; construct "point on object"; label as B.

Select points A and B; construct "segment".

Choose this segment and point B; construct "perpendicular line".

"If a line is to a circle, then it is to the

radius drawn to the point of tangency."

Construct another point on the circle; label as C. Choose points A and C; construct "segment".

Choose this segment and point C; construct "perpendicular line".

Move point C around the circle until the two tangent lines intersect.

Choose the two lines and construct "point at intersection"; label as D.

Measure distance: BD= CD =

Move point C around the circle and observe the distance measures of BD and CD.

"If two segments from the same exteriod point are tangent to a circle, then they are

."

Construct segment AD. Triangle DAB is a triangle.

Make AB = 3 (by dragging point z around). Make BD = 4 (by dragging B).

Using the Pythagorean Theorem, find the length of AD. (Show your work)

Construct Circle A. Hide the child.

Construct three points on the circle, B, C, and D. Construct acute angle BCD.

=

Arc BD =

Note: To measure a minor arc angle, select and the two points that define the arc. Then select "measure", "arc angle". To measure a major angle, you must first construct a new point X on the circle between the two points that define the arc. To find the measure of the arc, select the circle and the three points, making sure that X is in the middle of the three. Then select "measure", "arc angle".

Angle BCD is an angle and is equal to of it arc .

Move point B around until the angle is obtuse.

=

Intercepted arc " " = .

Delete segments CD and BC. Construct a point between points D and B, where C is not labeled; label as E.

Construct segments BD and CE. Make Arc BC = 60 and Arc DE = 140.

Choose each chord and construct "point at intersection"; label as F.

DEF = BFC =

Arc BC and Arc DE are the intercepted arcs for these angles.

Arc BC + Arc DE = . Compare the angle measures with the sum of the intercepted arc measures.

"If two chords intersect in the interior of a circle, then the measure of an angle formed is

the sum of the measures of the arcs intercepted by the angle and its vertical angle".

Hide segments BD, CE, and point F. Construct "lines" BE and CD. Move point B around until the lines intersect outside the circle; label the intersection point G.

Arc ED = Arc BC = |Arc ED - Arc BC| = BGC =

Compare the last two measures.

Case One: If two secants intersect in the exterior of a circle, then the measure of the angle formed is

of the positive difference of the measures as of the intercepted arcs.

Move point D on top of point C. Make sure Arc ED is a minor arc by moving point E around.

Arc ED = Arc BC = |Arc ED - Arc BC| = BGC =

Compare the last two measures.

Case Two: If a secant and a tangent intersect in the exterior of a circle, then the measure of the angle formed is of the positive difference of the measures of the intercepted arcs.

Move point E on top of point B. Make sure that point G remains visible. Construct a point F between points E/B and D/C in order to measure the major Arc EFD.

Arc ED = Arc BC = |Arc ED - Arc BC| = BGC =

Case Three: If two tangents intersect in the exterior of a circle, then the measure of the angle formed is of the positive difference of the measures of the intercepted arcs.