The numbernsatisfies the equation√80 +√125 =√n. What is the value ofn?(A)205(B)305(C)405(D)500(E)10002010 AMC 10 B, Problem #6—2010 AMC 12 B, Problem #4—“Note that√80 = 4√5and√125 = 5√5.”SolutionAnswer (C):Simplifying gives√80 +√125 = 4√5 + 5√5 = 9√5 =√405. Hencen= 405.Diﬃculty:Medium-easyNCTM Standard:Number and Operations Standard for Grades 9–12: understand numbers, waysof representing numbers, relationships among numbers, and number systems.
The square ofxis 16 larger than the square ofy, andxis 2 larger thany. What is the productxy?2010 AMC 10 B, Problem #7—“Note thatx2−y2= (x−y)(x+y) = 16.”Diﬃculty:Medium-easyNCTMStandard:Algebra Standard for Grades 9–12:write equivalent forms of equations,inequalities, and systems of equations and solve them with ﬂuency-mentally or with paper andpencil in simple cases and using technology in all cases.
The area of triangleEBDis one third of the area of3–4–5triangleABC. SegmentDEis perpendicularto segmentAB. What isBD?ABCDE345(A)43(B)√5(C)94(D)4√33(E)522010 AMC 10 B, Problem #8—“The area of△ABCis12·3·4 = 6.”Standard:Geometry Standard for Grades 9–12:analyze properties and determineattributes of two- and three-dimensional objects.
If a rectangle’s width is increased by 3, the rectanglebecomes a square and the area of the rectangle isincreased by 54.What is the area of the originalrectangle?(A)90(B)150(C)180(D)225(E)2702010 AMC 10 B, Problem #9—“Letxandybe the width and length of the originalrectangle, respectively. Form an equation to show thedifference in areas.”Standard:Geometry Standard for Grades 9–12:analyze properties and determineattributes of two- and three-dimensional objects.