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question:The diagonal of a square is sqrt{2} inches long. How many inches long is the side of the square?
answer:Since a diagonal of a square breaks up the square into two 45-45-90 triangles, a diagonal is sqrt{2} times longer than a side. Thus, the length of a side of the square is sqrt{2}/sqrt{2}=boxed{1} inch.
question:Four points B, A, E, and L are on a straight line, as shown. The point G is off the line so that angle BAG = 120^circ and angle GEL = 80^circ. If the reflex angle at G is x^circ, then what does x equal? [asy] draw((0,0)--(30,0),black+linewidth(1)); draw((10,0)--(17,20)--(15,0),black+linewidth(1)); draw((17,16)..(21,20)..(17,24)..(13,20)..(14.668,16.75),black+linewidth(1)); draw((17,16)..(21,20)..(17,24)..(13,20)..(14.668,16.75),Arrows); label("B",(0,0),S); label("A",(10,0),S); label("E",(15,0),S); label("L",(30,0),S); label("G",(17,20),N); label("120^circ",(10,0),NW); label("80^circ",(15,0),NE); label("x^circ",(21,20),E); [/asy]
answer:Since the sum of the angles at any point on a line is 180^circ, then we find that begin{align*} angle GAE &= 180^circ - 120^circ = 60^circ, angle GEA &= 180^circ - 80^circ = 100^circ. end{align*} [asy] draw((0,0)--(30,0),black+linewidth(1)); draw((10,0)--(17,20)--(15,0),black+linewidth(1)); draw((17,16)..(21,20)..(17,24)..(13,20)..(14.668,16.75),black+linewidth(1)); draw((17,16)..(21,20)..(17,24)..(13,20)..(14.668,16.75),Arrows); label("B",(0,0),S); label("A",(10,0),S); label("E",(15,0),S); label("L",(30,0),S); label("G",(17,20),N); label("120^circ",(10,0),NW); label("80^circ",(15,0),NE); label("x^circ",(21,20),E); draw((11,5.5)--(11.5,0.5),black+linewidth(1)); draw((11,5.5)--(11.5,0.5),EndArrow); draw((13,-4)--(14,1),black+linewidth(1)); draw((13,-4)--(14,1),EndArrow); label("60^circ",(11,5.5),N); label("100^circ",(13,-4),S); [/asy] Since the sum of the angles in a triangle is 180^circ, we have begin{align*} angle AGE &=180^circ - angle GAE - angle GEA &= 180^circ - 60^circ - 100^circ &= 20^circ. end{align*} Since angle AGE=20^circ, then the reflex angle at G is 360^circ - 20^circ = 340^circ. Therefore, x=boxed{340}.
question:Which of the following statements are true? A. 3 is a factor of 18. B. 17 is a divisor of 187 but not of 52. C. 24 is neither a divisor of 72 nor 67. D. 13 is a divisor of 26 but not of 52. E. 8 is a factor of 160. Write your answer with the letters in alphabetical order, separated by commas. For example, if you think all five are true, you should type "A,B,C,D,E" (without the quotes).
answer:A. Since 18=3cdot 6, there is an integer n such that 18=3cdot n. Therefore, by definition of factor, 3 is a factor of 18 and statement A is true. B. We can list the divisors of 187. They are 1, 11, 17, and 187. Therefore, 17 is a divisor of 187. We can also list the divisors of 52. They are 1, 2, 4, 13, 26, and 52. Therefore, 17 is not a divisor of 52 and statement B is true. C. Since 72=24cdot 3, there is an integer n such that 72=24cdot n. Therefore, by definition of divisor, 24 is a divisor of 72 and statement C is false. D. We already listed the divisors of 52 for statement B. Since 13 was one of them, 13 is a divisor of 52, and statement D is false. E. We can list the factors of 160. They are 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, and 160. Therefore, 8 is a factor of 160, and statement E is true. Therefore, the statements that are true are boxed{text{A,B,E}}.
question:Simplify sqrt{288}.
answer:We have: sqrt{288}=sqrt{144cdot 2}=boxed{12sqrt{2}}.