{"id":342,"date":"2022-04-13T10:44:16","date_gmt":"2022-04-13T10:44:16","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=342"},"modified":"2025-01-05T10:33:33","modified_gmt":"2025-01-05T10:33:33","slug":"r-6-arcs-and-chords","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/r-6-arcs-and-chords\/","title":{"rendered":"R.6 Arcs and chords"},"content":{"rendered":"\n

Arcs and chords<\/strong><\/h2>\n\n\n\n

Key Notes :<\/p>\n\n\n\n

Arclength<\/strong> is the distance between two points along a section of a curve or circle.<\/p>\n\n\n

\n
\"A<\/figure><\/div>\n\n\n

Here is a formula for arc length:<\/p>\n\n\n\n

\ud835\udcc1<\/strong>= m\/360 . C<\/p>\n\n\n\n

In the formula, \ud835\udcc1<\/strong> is the arc length, m is the degree measure of an arc (or the central angle that intercepts the arc), and C is the circumference of the circle.<\/p>\n\n\n\n

\ud83d\udd14 Tip<\/strong><\/p>\n\n\n\n

You can also write this formula as a proportion<\/a> where each ratio relates the arc to the full circle:<\/p>\n\n\n\n

\ud835\udcc1<\/strong>\/C = m\/360 <\/p>\n\n\n\n

The ratio \ud835\udcc1<\/strong>\/C compares the arc length to the circumference of the circle. The ratio m\/360 compares the degree measure of the arc to the degree measure of a full circle.<\/p>\n\n\n\n

Finding arc length<\/h4>\n\n\n\n

Let’s try it! The radius of the circle below is 8 feet. Find the length of a 120\u00b0 arc.<\/p>\n\n\n

\n
\"A<\/figure><\/div>\n\n\n

To find the arc’s length, you’ll need to use the arc’s measure and the circle’s circumference. Find the circle’s circumference using the formula C=2\u200b\ud835\udf0br, where r is the radius.<\/p>\n\n\n\n

C=2\u200b\ud835\udf0br<\/p>\n\n\n\n

= 2 . \u200b\ud835\udf0b . 8 Plug in r=8.<\/p>\n\n\n\n

= 16\u200b\ud835\udf0b Simplify.<\/p>\n\n\n\n

The circumference is 16\u200b\ud835\udf0b feet.<\/p>\n\n\n\n

Now, find the length of the arc.<\/p>\n\n\n\n

\ud835\udcc1= m \/ 360 . C<\/p>\n\n\n\n

= 120 \/ 360 . 16\u200b\ud835\udf0b Plug in m=120 and C=16\u200b\ud835\udf0b.<\/p>\n\n\n\n

= 16\u200b\ud835\udf0b \/ 3 Simplify.<\/p>\n\n\n\n

So, the length of the arc is 16\u200b\ud835\udf0b \/ 3 feet <\/p>\n\n\n\n

Finding arc measures<\/h4>\n\n\n\n

To find the measure of an arc, you can use the arc’s length and the circle’s circumference.<\/p>\n\n\n\n

Let’s try it! The diameter of the circle below is 12 inches. Find the measure, in degrees, of an arc that is 2\u200b\ud835\udf0b inches long.<\/p>\n\n\n

\n
\"A<\/figure><\/div>\n\n\n

To find the arc’s measure in degrees, you’ll need to use the arc’s length and the circle’s circumference. Find the circle’s circumference using the formula C=\u200b\ud835\udf0bd, where d is diameter.<\/p>\n\n\n\n

C=\u200b\ud835\udf0bd<\/p>\n\n\n\n

= 12\u200b\ud835\udf0b Plug in d=12.<\/p>\n\n\n\n

The circumference is 12\u200b\ud835\udf0b inches.<\/p>\n\n\n\n

Now, find the measure of the arc.<\/p>\n\n\n\n

\ud835\udcc1= m \/ 360 . C<\/p>\n\n\n\n

2\u200b\ud835\udf0b= m \/ 360 . 12\u200b\ud835\udf0b Plug in \ud835\udcc1=2\u200b\ud835\udf0b and C=12\u200b\ud835\udf0b<\/p>\n\n\n\n

2\u200b\ud835\udf0b . 360 \/ 12\u200b\ud835\udf0b = m Multiply both sides by <\/em>360 \/ 12\u200b\ud835\udf0b<\/p>\n\n\n\n

60= m Simplify<\/p>\n\n\n\n

So, the measure of the arc is 60\u00b0.<\/p>\n\n\n\n

Arc length and radians<\/h3>\n\n\n\n

You can also find arc length when the arc or the central angle is measured in radians. Here is a formula for arc length:<\/p>\n\n\n\n

\ud835\udcc1=<\/strong>r\u200b\ud835\udf03<\/p>\n\n\n\n

In the formula, \ud835\udcc1 is the arc length, r is the radius of the circle, and \u200b\ud835\udf03 is the radian measure of the arc (or the central angle that intercepts the arc).<\/p>\n\n\n\n

\ud83d\udd14 Tip<\/strong><\/p>\n\n\n\n

When using this formula to solve problems, you may be asked about an arc that subtends<\/strong> a given angle. This means that the endpoints of the arc are the points where the angle intersects the circle.<\/p>\n\n\n\n

For example, in the circle below, BC subtends \u2220BAC<\/p>\n\n\n\n

\n
\"Circle<\/figure><\/div><\/div><\/div>\n\n\n\n

Finding arc length<\/h4>\n\n\n\n

Let’s try it! The radius of the circle below is 12 centimeters. Find the length of an arc that subtends an angle of \u200b\ud835\udf0b\/2 radians.<\/p>\n\n\n

\n
\"A<\/figure><\/div>\n\n\n

You can use the central angle’s measure and the circle’s radius to find the arc’s length.<\/p>\n\n\n\n

\ud835\udcc1=r\u200b\ud835\udf03<\/p>\n\n\n\n

=12 . \ud835\udf0b\/2 Plug in r=12 and \u200b\ud835\udf03= \ud835\udf0b\/2<\/p>\n\n\n\n

= 6\u200b\ud835\udf0b Simplify.<\/p>\n\n\n\n

So, the length of the arc is 6\u200b\ud835\udf0b centimeters.<\/p>\n\n\n\n

Finding arc measures<\/h4>\n\n\n\n

You can use an arc’s length and the circle’s radius to find an arc’s measure in radians.<\/p>\n\n\n\n

Let’s try it! The radius of the circle below is 3 meters. Find the measure, in radians, of an arc that is 5\u200b\ud835\udf0b\/2  meters long.<\/p>\n\n\n

\n
\"A<\/figure><\/div>\n\n\n

Find the measure of the arc using the arc’s length and the circle’s radius.<\/p>\n\n\n\n

\ud835\udcc1=r\u200b\ud835\udf03<\/p>\n\n\n\n

5\u200b\ud835\udf0b\/2 = 3\u200b\ud835\udf03 Plug in \ud835\udcc1= 5\u200b\ud835\udf0b\/2 and r = 3<\/p>\n\n\n\n

5\u200b\ud835\udf0b\/6 = \u200b\ud835\udf03 Divide both sides by 3.<\/p>\n\n\n\n

So, the measure of the arc is  5\u200b\ud835\udf0b\/6 radians<\/p>\n\n\n\n

\ud83d\udd14 Fun Fact<\/strong><\/p>\n\n\n\n

The formula \ud835\udcc1=r\ud835\udf03 comes from the definition of radians. The radian measure of a central angle in a circle is \ud835\udf03=\ud835\udcc1\/r, where \ud835\udcc1 is the length of the arc that the angle intercepts, and r is the radius of the circle. You can solve that equation for \ud835\udcc1 to get the formula.<\/p>\n\n\n\n

\ud835\udf03=\ud835\udcc1\/r Definition of radian<\/p>\n\n\n\n

r\u200b\ud835\udf03= \ud835\udcc1 Multiply both sides by r.<\/p>\n\n\n\n

Learn with an example<\/strong><\/p>\n\n\n\n

\n
\n

What is the length of  ST ?<\/strong><\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

ST = _______\u00b0<\/p>\n<\/div><\/div>\n\n\n\n

RU and ST are arcs in a circle, and their corresponding chords are congruent. So, RU is congruent to ST.<\/p>\n\n\n\n

ST=RU=77\u00b0.<\/p>\n<\/div><\/div>\n\n\n\n

\n
\n

What is the measure of CD ?<\/strong><\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

CD = ________\u00b0<\/p>\n<\/div><\/div>\n\n\n\n

BE and CD are arcs in a circle, and their corresponding chords are congruent. So, BE is congruent to CD.<\/p>\n\n\n\n

CD = BE = 68\u00b0.<\/p>\n<\/div><\/div>\n\n\n\n

\n
\n

What is the measure of  HI ?<\/strong><\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

HI = _______\u00b0<\/p>\n<\/div><\/div>\n\n\n\n

FG and HI are arcs in a circle, and their corresponding chords are congruent. So, FG is congruent to HI.<\/p>\n\n\n\n

HI=FG=55\u00b0.<\/p>\n<\/div><\/div>\n\n\n\n

let’s practice!<\/p>\n\n\n\n

\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Arcs and chords Key Notes : Arclength is the distance between two points along a section of a curve or circle. Here is a formula for arc length: \ud835\udcc1= m\/360 . C In the formula, \ud835\udcc1 is the arc length, m is the degree measure of an arc (or the central angle that intercepts the arc), and C is the circumference of the circle. \ud83d\udd14 Tip You can also write this formula as a proportion where each ratio relates the arc to the full circle: \ud835\udcc1\/C =Continue reading “R.6 Arcs and chords”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-342","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/342","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=342"}],"version-history":[{"count":13,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/342\/revisions"}],"predecessor-version":[{"id":17369,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/342\/revisions\/17369"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=342"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}