Anonymous
Anonymous asked in Science & MathematicsMathematics · 9 years ago

# Circle Geometry: Find Different Angles?

Inside circle O, tangent PW and secant line PST are drawn. P is located outside the circle. Chord WA is parallel to chord ST. Chords AS and WT intersect at point B in the circle.

If the measure of arch WA : measure of arch AT : measure of arch ST = 1 : 3 : 5, find the following angles:

(a) measure of angle TBS

(b) measure of angle TWP

(c) measure of angle WPT

(d) measure of angle ASP

Relevance

Let's first find the measures of the arcs.

Since chords WA and ST are parallel, alternate interior angles STW and TWA are equal.

Note that the equal angles STW and TWA are inscribed angles that intercept arcs SW and AT, respectively, so since the measure of an inscribed angle is half the measure of its intercepted arc, arcs SW and TA are equal.

Note that the sum of the measures of WA, AT, ST, and SW add to 360 degrees. Since the ratio of the measures of WA, AT, ST is 1:3:5 (in this order), and arcs SW and AT are equal, we find that

measure of arc WA = 360(1)/(1 + 3 + 5 + 3) = 30 degrees.

measure of arc AT = 360(3)/(1 + 3 + 5 + 3) = 90 degrees.

measure of arc ST = 360(5)/(1 + 3 + 5 + 3) = 150 degrees.

measure of arc SW = 360(3)/(1 + 3 + 5 + 3) = 90 degrees.

Now recall that the measure of an angle at a vertex inside the circle is half the sum of the measures of the two intercepted arcs; the measure of an angle at a vertex outside the circle is half the difference of the measures of the two intercepted arcs; the measure of an inscribed angle (or an angle formed by a tangent and a chord with vertex at the point of tangency) is half the measure of the intercepted arc.

(a) measure of angle TBS

= (1/2)(measure of arc WA + measure of arc ST)

= (1/2)(30 + 150)

= 90 degrees

(b) measure of angle TWP

= (1/2)(measure of arc TSW)

= (1/2)(measure of arc ST + measure of arc SW)

= (1/2)(150 + 90)

= 120 degrees

(c) measure of angle WPT

= (1/2)(measure of arc WAT - measure of arc SW)

= (1/2)(measure of arc WA + measure of arc AT - measure of arc SW)

= (1/2)(30 + 90 - 90)

= 15 degrees

(d) measure of angle ASP

= 180 - measure of angle AST

= 180 - (1/2)(measure of arc AT)

= 180 - (1/2)(90)

= 180 - 45

= 135 degrees

Lord bless you today!

• Login to reply the answers
• A good picture makes all the difference. If you weren't given one, I suggest that you try to draw one.

The first thing you should notice is that since WA || ST, that means that the measure of arc AT is equal to the measure of arc WS. So the ratios of the four arcs that make up the circumference are 1:3:5:3. This makes for 12 total parts, so the measures of the arcs are:

mWA = 30

mAT = 90

mTS = 150

mSW = 90

Once you have these, the rest fall into place.

a) The measure of angle TBS is the average of the arcs intecercepted by that pair of vertical angles. Since the arcs have measures of 30 and 150, the measure of angle TBS is 90,

b) First note that the measure of angle WTP is half of its intercepted arc, so mWTP = 45. Note that the measure of angle WPT is equal to half the difference between the two arcs it creates, so (120-90)/2 = 15. Angle TWP is the third angle of a triangle with the other two angle measures, so 120.

• Login to reply the answers