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This problem is like a vocabulary quiz - it sounds complicated, but I bet
if you read it carefully and draw a good picture, you can get it.
A chord of a circle is the hypotenuse of an isosceles right triangle whose legs are radii of the circle. The length of the chord is 6 times the square root of 2. What is the length of the minor arc subtended by the chord? SolutionsI like this problem because while it is not difficult, there are a lot of little details you need to get right in order to get to the end, which makes it a lot harder. 103 students got it right, and 30 got it wrong. It does help to know all of the vocabulary because in order to draw a good picture to help you figure it out, you need to know what is going on. You don't necessarily need to know the meaning of the word "subtend," because if you read the rest of the problem carefully enough, there might be enough clues to figure it out. However, looking up the terms you don't understand never hurts, and you might even learn something! Whitney, Lauren, and Katy from Germantown Academy did just that, and wrote a nice solution that you can read below. I've also included the solution of Kristen Allegue from the Ethel Walker School. A good picture always helps in a problem like this, and I have included the solution from Rashida Fisher of Highland Park Senior High School because it makes it very clear what is going on and what we are looking for. A number of students got the problem wrong because they found the "measure" of the arc, which is 90. There is a difference between "measure" and "length" - you need to read a problem carefully to see what is being asked for. I suspect most of those students had just learned the measure of an arc in class, and so they were a little too focused on that concept. There was a great push this week toward over-rounding. This is mostly because students didn't want to use 6sqrt2 in their problem, but had to change it to its decimal equivalent. Then they rounded that, and when you square it to find the radius, it doesn't come out to 6, but something really close. Here's a tip: when you have numbers you don't like, like fractions or radicals or whatever, write out your equation with the given numbers. You might find that they will work out really nicely without a calculator! 6sqrt2 comes out to a nice neat number (72) that is much better than 72.25 or whatever you might get when you do it the other way. Now, our spelling lesson for the week. Once again, It's "pi," not "pie." Pie is a food product that is usually shaped like a circle, so we get confused. Pi is a Greek letter that we use to denote that funky number. Also try to proofread your solution, and look up any words you might not be sure of, such as Pythagoras, Pythagorean, and hypotenuse. A list of all the people who got this problem right follows the highlighted solutions below, and most of the solutions are also available.
From: Whitney and Lauren and Katy
Grade: 9
School: Germantown Academy, Fort Washington, Pennsylvania
The first thing we did was look up the terms in the problem which we did not
know. Once we knew all of the terms, we drew a diagram. We realized that the
two legs of the triangle were congruent. We figured out the length of the legs
of the triangle by using the equation l^2 + l^2= h^2. This came out as each leg
being 6. This means that 6 is the radius of the circle. We thought the next
step would be to find the circumfrence of the circle. To find the circumfrence
of the circle we used the equation 2(pi)r=Circumfrence. When we did this we got
that the circumfrence was 37.7 . We then figured out that the minor arc is 1/4
of the circumfrence of the circle because the two legs are both radii of the
circle and they are at a right angle. This divides the circle and the part that
is the minor arc is 1/4 of the total circumfrence. The circumfrence divided by
4 equals 9.42 the length of the minor arc is 9.42.
From: Kristen Allegue
Grade: 9
School: Ethel Walker School, Simsbury, Connecticut
The length of the minor arch substended by the chord is 3*Pi.
In order to come across my answer, I began to take the problem a
part by drawing a diagram. I drew a circle and I placed the
vertex of the right angle in the center of the circle. The
legs of the triangle are the radius of the circle. So we can
find the radius of the circle by using Pathagorean Theorem,
which in our case is r^2 + r^2=36*2 (since the legth of the chord
is six times the square root of two). Solving this equation, I get
2*r^2=72, or r^2=36. Hence, r=6. Now, the length of the minor arc
is 1/4 of the length of the circummference of the circle.
Therefore, the length of the minor arch is 1/4*2*6*Pi=3*Pi.
From: Rashida Fisher
Grade: 9
School: Highland Park Senior High School, St. Paul, Minnesota
Subject: Oct 31 POW
Rashida Fisher, Grade 9, Geometry IB
Highland Park Senior High School, (612) 293-8940
www.stpaul.k12.mn.us/hphs/highland.html
October 27 - 31 Problem of the Week
The following students submitted correct solutions this week:James Tong, Grade 11, Klein High School, Klein, TexasChristoph Mentz, Grade 13, Gymnasium Vaterstetten, Germany Gordon Bockus Jr., Grade Freshman, Eastern Oklahoma State College, Wilburton, Oklahoma Catherine Mangasi, Grade 12, Wilburton High School, Wilburton, Oklahoma Greg Schoppe, Grade 6, homeschooled Lunenburg, Vermont Chris Shaw, Grade 10, Middletown High School North, Middletown, New Jersey Tiffanie Lam, Grade 8, Sequoia Middle School, Pleasant Hill, California Caitlin Brennan, Grade 9, Rufus King High School, Milwaukee, Wisconsin Jenny Lurie, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Will Marrs, Grade 9, Granada High School, Livermore, California Kelvin Liang, Grade 8, Issaquah Middle School, Issaquah, Washington Evan Bruhn and Will Burgess, Grade 9, West Point High School, West Point, Virginia Alex Glocer, Grade 12, Cape Coral High School, Cape Coral, Florida Anthony Nelson, Grade 12, Valley City High School, Valley City, North Dakota Alexandra Sowa, Grade 9, Archmere Academy, Wilmington, Delaware Candice Hebden, Grade 10, William Penn Charter School, Philadelphia, Pennsylvania Angela Ann Holland, Grade 9, Skyview High School, Vancouver, Washington Avrum Tilman, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Jennifer Liang, Grade 8, Odle Middle School, Bellevue, Washington Sean Taylor, Grade 10, William Penn Charter School, Philadelphia, Pennsylvania Whitney and Lauren and Katy, Grade 9, Germantown Academy, Fort Washington, Pennsylvania Anna Wu, Grade 10, Monte Sant' Angelo Mercy College, Sydney, Australia Bryan Barnett, Grade 9, Skyview High School, Vancouver, Washington Wassia Khaja, Grade 8, Albright Middle School, Houston, Texas Nessie G., Grade , Dubai, United Arab Emirates Mrs. Sharma's Pd C, Grade 11,12, Manchester Memorial High School, Manchester, New Hampshire Chaim Bloom, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Grant Braswell, Grade 8, Georgetown Day School, Washington, DC David Zax, Grade 8, Georgetown Day School, Washington, DC Beth McCabe and Amanda Woodruff, Grade 11, Wilburton High School, Wilburton, Oklahoma Kamila Sikora, Grade 9, Smoky Hill High School, Aurora, Colorado Ashley Tierney, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania Christina Tart and Jennie Doss, Grade 9, Franklin County High School, Rocky Mount, Virginia Robin Thornburg, Grade , Shelby County High School, Columbiana, Alabama Denise Dotson and Jessica Sink, Grade 9, Franklin County High School, Rocky Mount, Virginia Sara FitzSimmons and Jane Milton, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania Kanth Sishtla, Grade 8, Jefferson Junior High School, Woodridge, Illinois Jon Gantman, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Arielle Cohen, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Hannah Margoles, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Isaac Matthias, Grade 10, William Penn Charter School, Philadelphia, Pennsylvania Julia Le, Grade 11, Minnechaug Regional High School, Wilbraham, Massachusetts Alex Morgovsky, Grade 11, Akiba Hebrew Academy, Merion, Pennsylvania Sangmin Lee, Grade 11, J.P. Taravella School, Coral Springs, Florida Frannie Laks, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Sarah Molouki and Laura Cowen, Grade 10, Holton-Arms School, Bethesda, Maryland Daniel Branscombe, Grade Freshman Actuarial Science major, Drake University, Des Moines, Iowa Jenny Kaplan, Grade 7, Castilleja Middle School, Palo Alto, California Mark Garvin, Grade 12, Chapel Hill High School, Chapel Hill, North Carolina Chester Chan, Grade 6, Odle Middle School, Bellevue, Washington Ashley Monroe, Grade 9, Casady School, Oklahoma City, Oklahoma Jeffrey Wong, Grade 8, St. Ann School, Willaston, Massachusetts Sorin Ionescu, Grade 8, Ecole Secondaire Dorval, Quebec, Canada Josh Osborn, Grade 9, Skyview High School, Vancouver, Washington Sonia Teas, Grade 10, Oak Park and River Forest High School, Oak Park, Illinois Lev Navarre, Grade 6, Odle Middle School, Bellevue, Washington Tessa Goodhew, Grade 10, Lakeside School, Seattle, Washington Andrew Davis, Grade 9, Skyview High School, Vancouver, Washington Katie B. Anthony, Grade 9, Casady School, Oklahoma City, Oklahoma Tracy Steed, Grade 12, Wilburton High School, Wilburton, Oklahoma Leslie Scott, Grade 9, Casady School, Oklahoma City, Oklahoma Kristen Allegue, Grade 9, Ethel Walker School, Simsbury, Connecticut Josh Kinney, Grade 7, Chillicothe Junior High School, Chillicothe, Illinois Phillip Jeffery, Grade , Redmond High School, Redmond, Oregon Joanna Frankel, Grade 9, Germantown Academy, Fort Washington, Pennsylvania Joshua Zucker, Grade tutor, Stanford University, Palo Alto, California William Christianson, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Emily Buzicky, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Jason Chiu, Grade 9, Laramie Junior High School, Laramie, Wyoming Hunter Brooks, Grade 8, Camelot Academy, Durham, North Carolina Holly Black, Grade 7, Odle Middle School, Bellevue, Washington Alison Falkenhagen, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Brandon Gilchrist, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Thuy Nguyen, Grade 11, Highland Park Senior High School, St. Paul, Minnesota Kelsey Long, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Rashida Fisher, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Cooper Offenbecher, Grade , Lakeside School, Seattle, Washington Laura Roos, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania Rachel Toaff, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Doug Yoder, Grade 12, Highland Park Senior High School, St. Paul, Minnesota Jen Erhart, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Amy Tappe, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Kelly Flis, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Melanie Hudak and Amanda Gelik and Josh Sadowski and Mike B., Grade , Shaler Area High School, Pittsburgh, Pennsylvania Nick LeDonne and Emily Demich and Stacy Lynch and Chuck Krcelic, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Brian Bailey and Jill Bisceglia and Justin Dembowski and Niki Weber, Grade , Shaler Area High School, Pittsburgh, Pennsylvania John Kreke and Jill Filiponte and Jamie Jansen and Leigh Seserko, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Tony Kambic and Elisabeth Didomenico and Mary Richardson and Brian Fogle, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Emily Schuler and Jessica Schwartzmeier and Kaitlin Martorelli, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Alexander Chen, Grade 6, Odle Middle School, Bellevue, Washington Ben Yeckel, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Megan Bray, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania Matt Niederst, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Ken Rockot and Anthony Schwab and Eric Rastetter and Justin Hernandezicolapa, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania Pete and Leif, Grade , Redmond High School, Redmond, Oregon Allen Hsu, Grade , Nitschmann Middle School, Bethlehem, Pennsylvania Alex Chernyavsky, Grade , Akiba Hebrew Academy, Merion, Pennsylvania Alison Miller, Grade 6, homeschooled, Niskayuna, New York Emily Rozak and Melanie Seng, Grade 10, Germantown Academy, Fort Washington, Pennsylvania Lauren Rossi and Anne Hines, Grade 10, Germantown Academy, Fort Washington, Pennsylvania Thomas Kuo, Grade 10, Burroughs High School, Ridgecrest, California Andrew Cooledge, Grade 7, Odle Middle School, Bellevue, Washington Cory Campbell and Ashley Szlec and Clint Soose and Allen Faler, Grade , Shaler Area High School, Pittsburgh, Pennsylvania |
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To find the length of the red arc, find r.
To find r, use the Pythagorean Theorem.
r^2 + r^2 = (6 radical 2)^2
2r^2 = 72
r^2 = 36
r = 6
Use the formula for circumference to find the circumference of the circle.
C = 12(pi)
Divide the circumference by 4 because the arc is one-fourth of the circle (90
degrees out of 360 degrees). arc length = 12(pi)/4 = 3(pi).