stapel said:
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As a courtesy to viewers, here is the second exercise:
Take a circle with center O. On the circle, draw point A at the two-o'clock position, point G at the five-o'clock position, point D at the seven-o'clock position, and point B at the eight-o'clock position. (All positioning is approximate and should not be assume to be "to scale".) Draw secants AB and CD; label their crossing point as point E. Draw secant BG and extend BG to the right past G. Draw a tangent at A down to the extension of BG; label the intersection point as point P.
Note: AB should pass through center point O; the intersection of CD and BG is called point F.
Given: The measure of arc CA is 70; the measure of arc DG is 90; the measure of angle CEA is 40°.
a) Find the measure of arc CB.
b) Find the measure of arc BD.
c) Find the measure of angle APB.
d) Find the measure of angle PAB.
e) Find the measure of angle ABG.
Thanks, Eliz, for the "verbal description" of the diagram....unfortunately, you didn't mention the location of C, which is perhaps at the 12 o'clock position.
Anyway, for GWS, this is a relatively straightforward problem which requires you to know a few things about circles and angles:
1) The measure of a semicircle is 180. You will use this to answer part (a). You are given the measure of arc CA, and you know that the measures of arc CA and arc CB must add up to 180, because arc BCA is a semicircle.
2) The measure of an angle formed by two chords or secants which intersect
inside a circle is half the
sum of the intercepted arcs. For part (b),
m/_CEA = (1/2)(arc AC + arc BD)
You know the measures of /_CEA and arc AC....some basic algebra should get you the measure of arc BD.
3) The measure of an angle formed by two secants, two tangents, or a secant and a tangent which intersect
outside a circle is half the
difference of the arcs intercepted by the angle. For part (c),
m/_APB = (1/2)(arc ACB - arc AG)
You know that ACB is a semicircle with a degree measure of 180. You've found the measure of arc BD in the previous exercise, and you know that the measure of arc DG is 90. You should be able easily to determine the measure of arc AG.
4) A radius drawn to the point of tangency is
perpendicular to the tangent. What kind of angles are formed by perpendicular lines? This should make answering part (d) real simple.
5) An inscribed angle is formed by two chords (or secants) which intersect
on the circle. The measure of an inscribed angle is half of its intercepted arc. For part (e), angle ABP is an inscribed angle. It intercepts arc AG. You have previously found the measure of arc AG.
m/_ABP = (1/2)(measure of arc AG)
I hope this helps you.
