A chord of a circle is the hypotenuse of an isosceles right triangle whose legs are radii of the circle. The radius of the circle is 6 times the square root of 2. What is the length of the minor arc subtended by the chord?What if it were an equilateral triangle instead of just isosceles?
What if I told you that the non-base angle (the angle at the vertex at the center of the circle) was x degrees? Then what would the answer be?
This problem has lots of parts, but I think if you really understand the first part, the second two won't be too bad. Make sure you really explain your answer!
Annie says: Solutions
A pretty good job this week. The majority of the incorrect solutions were only missing the third part - it is sometimes hard to generalize something even though you used it correctly to solve two other parts. One common mistake was to get the third part backwards, that is, dividing instead of multiplying by x/360, even after doing exactly the first thing in the first two parts. You need to look very carefully at what you did in the specific cases.
Several people used a formula for arc length that requires the length of the chord and the angle. It works okay for the first two parts, but I think it makes the third part harder.
For accuracy, leave everything in terms of pi and sqrt2 until the very end. You might also find that a lot of stuff cancels out before you're done, and to me, the fewer numbers, the better! In fact, with a problem like this, I like to see the answers in terms of pi and sqrt2 and as just numbers. It makes the answer easier to follow and connects it with the circle and the original radius.
This problem offers an opportunity to work on writing good equations. Explain carefully what you are going to do and where the numbers come from, but when it comes to presenting the numbers, write some equations.
I've highlighted two solutions this week. Thomas Bereknyei from Hillview Junior High gave a simple explanation, and Greg Moore and Shawn Phelen from Shaler Area High School pretty much explained it perfectly.
A list of all the people who got this problem right and most of the solutions are also available; there are a number of good solutions in the full list, so be sure you read them over.
Thomas Bereknyei Grade: 8 School: Hillview Junior High School, Pittsburgh, California 1. The answer to the first question is:13.32 Here is why: Because the non-base angle is 90 degrees (right triangle) the arc would be 1/(360/90) = 1/4 of the circle's circumference. The circle's circumference = 2r Pi = 2 * 6* SQRT(2) * 3.14 = 53.28. The length of the minor arc is: 53.28 / 4 = 13.32 2. Instead of dividing by 4, now we divide by 360/60 = 6 The length then of the minor arc is: 53.28 / 6 = 8.88 3. We already know the circumference is 53.28. 360 / x is what you would divide from 53.28 so the length of the minor arc is: 53.28 * x ------------ in terms of x which is the angle 360
Greg Moore and Shawn Phelen Grade: 10 School: Shaler Area High School, Pittsburgh, Pennsylvania The base angle in the measurment of revolutions times the circumference of the circle is equal to the minor arc subtended by the chord formed by the triangle with the base angle as one of its vertices. Using this we can form an equation: x/360 = revolutions of base angle where "x" is equal to degrees of angle. 2 * r * Pi = circumference of the circle x x*r*Pi the length of the minor arc --- * 2*r*Pi = -------- = where "x" is equal to the 360 180 base angle in degrees In the problem "r" is equal to 6 * 2^.5. In the problem "x" is equal to 90 degrees, 60 degrees and x degrees. answer 1 = 3 * 2^.5 * Pi = 13.328649 answer 2 = 2 * 2^.5 * Pi = 8.8857659 answer 3 = 2^.5*Pi*x --------- 30
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