finding measures of the arcs and angles | Mathematics | Page 1

icy spade May 9, 2023, 10:06 PM

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pls help

coarse orbitBOT May 9, 2023, 10:06 PM

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icy spade May 9, 2023, 10:07 PM

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measures of the arcs and angles***

coarse orbitBOT May 9, 2023, 10:07 PM

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finding measures of the arcs and angles

mental zinc May 9, 2023, 10:12 PM

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Hey there!

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One way we can find the measure of arc BD is by finding its corresponding central angle.

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This is because the measure of the central angle is congruent to the arc it intercepts/corresponds to.

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So which central angle intercepts arc BD? @icy spade

icy spade May 9, 2023, 10:15 PM

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60?

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😭

lucid delta May 10, 2023, 1:46 AM

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im pretty sure mBD = 120 degrees, mCED = 180 degrees

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is this in degrees or radians?

mental zinc May 10, 2023, 2:30 AM

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icy spade 60?

No. angle BOD is the central angle that intercepts arc BD.

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But look at angle COB and angle BOD.

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Notice how they lie next to each other on the diameter of the circle.

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Therefore, angle COB and angle BOD are supplementary angles.

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Hence: angle COB + angle BOD = 180

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60 + angle BOD = 180

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∴ angle BOD = 120

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The central angle is congruent to the arc it intercepts.

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angle BOD is a central angle that intercepts arc BD.

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Therefore, arc BD = 120 degrees.

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Do you understand? @icy spade

icy spade May 10, 2023, 2:32 AM

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Ohhh ya

mental zinc May 10, 2023, 2:32 AM

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Now look at arc CED.

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Notice how it is intercepted by angle COD.

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Segment COD cuts through the circle's center (O) and connects two endpoints that lies on the circumference of the circle (C & D).

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So what kind of part of a circle is segment COD?

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hint: ||chord, diameter, radius, secant, or tangent||

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@icy spade

icy spade May 10, 2023, 2:35 AM

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Secant

mental zinc May 10, 2023, 2:37 AM

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Hmm not quite. A secant is a line that cuts through two points that lies on the circle.

icy spade May 10, 2023, 2:37 AM

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Oh wait diameter

mental zinc May 10, 2023, 2:37 AM

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Right!

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segment COD is a diameter of the circle!

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the diameter of a circle is 180 degrees

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so this means that angle COD is 180 degrees

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since central angle COD intercepts arc CD:

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arc CD = 180 degrees

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do you get that?

icy spade May 10, 2023, 2:38 AM

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Ok ya

mental zinc May 10, 2023, 2:39 AM

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Now look at line AB and line CE.

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Notice how line AB intersects circle O at exactly one point, which is at point B

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Notice how line CE intersects circle O at exactly one point, which is at point C

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So what kind of parts of a circle are lines AB and CE?

mental zinc May 10, 2023, 2:40 AM

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mental zinc hint: ||chord, diameter, radius, secant, or tangent||

refer back to this hint if you are not sure

icy spade May 10, 2023, 2:40 AM

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Hm tanget

mental zinc May 10, 2023, 2:40 AM

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Correct!

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Line AB is tangent to circle O at point B, and line CE is tangent to circle O at point C

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The tangent line is perpendicular to the radius that contains the point of tangency

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So OB is a radius that contains point B. OC is a radius that contains point C.

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Therefore:

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Do you get it?

icy spade May 10, 2023, 2:42 AM

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Ya

mental zinc May 10, 2023, 2:42 AM

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Ok so now look at figure ABOC

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Notice how it has 4 sides

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This means that figure ABOC is a quadrilateral (a 4-sided shape)

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The sum of the interior angles of a quadrilateral is 360 degrees

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So this means that:

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angle COB + 90 + 90 + angle CAB = 360

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60 + 90 + 90 + angle CAB = 360

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From here, solve for angle CAB. Ping me when you get the answer.

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OH SORRY

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WAIT

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I DIDNT SEE THE PROBLEM LOL

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i thought we had to solve for angle CAB for some reason

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angle ABO = 90 degrees

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put that

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sorry

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idk why i saw angle CAB lol

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@icy spade

icy spade May 10, 2023, 2:44 AM

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😭😹ok

mental zinc May 10, 2023, 2:45 AM

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oh shit 🤦‍♀️

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omg sorry

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i just realised that CE is a secant

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not tangent

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omg sorry its like really late in the night for me

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pls forgive me 🙏

icy spade May 10, 2023, 2:46 AM

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No worries

mental zinc May 10, 2023, 2:46 AM

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Okay so now look at angle DCE.

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Notice how it is an inscribed angle.

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The inscribed angle is **half ** the measure of the arc it intercepts.

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So what arc does angle DCE intercept?

icy spade May 10, 2023, 2:47 AM

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120?

mental zinc May 10, 2023, 2:47 AM

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Nope

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This picture may help

icy spade May 10, 2023, 2:48 AM

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60

mental zinc May 10, 2023, 2:48 AM

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Nope angle DCE intercepts arc DE

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Because angle DCE is formed by chord CD and chord CE.

icy spade May 10, 2023, 2:49 AM

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Oh

mental zinc May 10, 2023, 2:49 AM

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chord CD touches the circle at points ** D and C**. chord CE touches the circle at points E and C

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you get it now?

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it may help to trace where the inscribed angle touches the circle

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to find the arc it intercepts

icy spade May 10, 2023, 2:49 AM

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Ok

mental zinc May 10, 2023, 2:49 AM

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So from here, you utilise the Inscribed Angle Theorem

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The inscribed angle theorem states that the inscribed angle is half the measure of the arc it intercepts

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So:

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angle DCE = 1/2(arc DE)

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angle DCE = 1/2(70)

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So what is angle DCE?

icy spade May 10, 2023, 2:51 AM

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35

mental zinc May 10, 2023, 2:51 AM

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Correct!

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Great job 👍

icy spade May 10, 2023, 2:51 AM

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Wooo

mental zinc May 10, 2023, 2:51 AM

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🥳

icy spade May 10, 2023, 2:52 AM

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Tysm 😭😭😭🧎‍♀️

mental zinc May 10, 2023, 2:52 AM

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of course! any time! ❤️ ❤️ ❤️

trim thicket May 10, 2023, 9:41 PM

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.solved

#finding measures of the arcs and angles