MATH QUESTION! ACT in 2 days! HELP

Test Preparation ACT Preparation
1

<p>In the figure below, line MR is a tangent to the circle at M. triangle NLM is inscribed in the circle, and the angle measures are as marked. Find Y.</p>

<p>Image is here : <a href="http://i52.tinypic.com/syvwj5.jpg%5B/url%5D"&gt;http://i52.tinypic.com/syvwj5.jpg&lt;/a&gt;&lt;/p>

<p>Please tell me step by step instructions on how to solve it and explain the concepts! its a Hard question according to ACT!</p>

<p>BTW: btw in the picture the far right vertex of triangle is 20 degrees and the angle beside Y is 75 degrees. </p>

<p>"N" is the top vertex of triangle, M is the far right vertex of triangle and L is the far left vertex of triangle.</p>

<hr>

<p>I should mention the correct answer according to Book:</p>

<p>Solution: The measure of arc LM is 150 degrees since the angle formed by tangent and secant measures half f its intercepted arc. </p>

<p>therefore m<LNM = 75 degrees (angle at circumference measures half its intercepted arc)</p>

<p>therefore: y = 85 (sum of angles in triangle is 180 degrees)</p>

2

<p>N= 90
L= 20</p>

<p>Thus,
Y= 70?</p>

<p>i am not sure if i’m correct.</p>

3

<p>iamanapp</p>

<p>you are wrong</p>

4

<p>

</p>

<p>you most probably wont get questions as difficult as this on the real test. (unless this is a Real ACT question, is it?)</p>

5

<p>its from a practice ACT book…</p>

6

<p>^ which one? just wondering…</p>

7

<p>Barrons ACT math and science workbook
<a href=“http://ecx.images-amazon.com/images/I/51dGj2EhGIL._BO2,204,203,200_PIsitb-sticker-arrow-click,TopRight,35,-76_AA300_SH20_OU01_.jpg[/url]”>http://ecx.images-amazon.com/images/I/51dGj2EhGIL._BO2,204,203,200_PIsitb-sticker-arrow-click,TopRight,35,-76_AA300_SH20_OU01_.jpg&lt;/a&gt;&lt;/p&gt;

<p>pg. pg 150 , ex 11</p>

8

<p>I have the same one. It’s a but harder than the real thing so use it to understand concepts and stuff but if a question seems over-the-top difficult, ignore it. It’s great in the sense that it prepares you well, but just don’t get freaked out if the questions don’t make sense. to practice, use the real ACT questions. good luck! :)</p>

<p>i hope someone more knowledgable than me answers your question. :P</p>

9

<p>Here’s the general outline of how to solve this problem (it’s pretty much how they explain it in the book):</p>

<ol>
<li><p>Recognize that the total number of degrees in a triangle is 180 - thus, angle “Y” must equal [180 - 20 - angle “LNM”].</p></li>
<li><p>The next step, then, is to solve for angle “LNM”. The only way to do this is to use the geometric theorems on angles, chords, and tangents. That sounds far more complicated than it is.</p></li>
</ol>

<p>Use this webpage as a base for reviewing these theorems (also a handy reference for many other subjects - explore!): [Circles</a> - angles formed by radii, chords, tangents, secants](<a href=“http://www.regentsprep.org/Regents/math/geometry/GP15/CircleAngles.htm]Circles”>http://www.regentsprep.org/Regents/math/geometry/GP15/CircleAngles.htm)</p>

<p>We’ll need to use theorem 3 first - the one involving a tangent and chord. Simply put, the measure of the angle formed by a chord (the line inside the circle) and the tangent (the line perpendicular to a point on the circle) equals 1/2 of the length of the arc in degrees. In the example on that website:</p>

<p>angle ABC = 1/2 of arc AB
angle ABC = 1/2 * 120 degrees
angle ABC = 60 degrees</p>

<p>We’re using the same concept in the practice problem, but in reverse. We have the angle - we need the arc length. So, using the same theory:</p>

<p>angle LMR = 1/2 of arc LM => divide both sides by 1/2 to get:
arc LM = 2 * angle LMR</p>

<p>Now solve for arc LM:</p>

<p>arc LM = 2 * angle LMR
arc LM = 2 * 75 degrees
arc LM = 150 degrees</p>

<p>So we now have the length of arc LM. We’re halfway to solving for angle LNM!</p>

<ol>
<li>We now have the length of arc LM, which we will now use to solve for angle LNM. Here’s how:</li>
</ol>

<p>Theorem number 2 on the website above describes the relationship between an “inscribed” angle (that is, an angle drawn “inside” of a circle) and the arc it intercepts (the part of the actual circle that it intersects). It’s a simple equation, identical to the one above actually (theorem 3 is a variation of theorem 2). Simply put, the measure of an angle inscribed in a circle is equal to 1/2 of the intercepted arc. In the example on that website:</p>

<p>angle ABC = 1/2 of arc AC
angle ABC = 1/2 * 100 degrees
angle ABC = 50 degrees</p>

<p>Now we use that same concept to find angle LNM, which intercepts arc LM (we have that measure!) - this time, it’s exactly the same as in that example. In the practice problem:</p>

<p>angle LNM = 1/2 of arc LM
angle LNM = 1/2 * 150 degrees
angle LNM = 75 degrees</p>

<p>And there we have it! Angle LNM is equal to 75 degrees; from here, the problem becomes simple algebra. Recall that we said earlier that:</p>

<p>angle Y = 180 - 20 - angle LNM</p>

<p>Now we can plug in angle LNM for angle Y!</p>

<p>angle Y = 180 - 20 - 75
angle Y = 160 - 75
angle Y = 85 degrees</p>

<p>And there’s the answer that the book gives! To summarize:</p>

<ol>
<li><p>Angle Y = 180 - 20 - Angle LNM</p></li>
<li><p>Arc LM = 2 * Angle LMR = 2 * 75 degrees = 150 degrees</p></li>
<li><p>Angle LNM = 1/2 * Arc LM = 1/2 * 150 degrees = 75 degrees</p></li>
<li><p>Angle Y = 180 - 20 - Angle LNM = 180 - 20 - 75 = 85 degrees</p></li>
</ol>

<p>Hope that helps! That website provides excellent review for almost if not all of the math on the ACT. Good luck!</p>