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Question 1210573: H is the orthocenter of acute triangle ABC and the extensions of AH, BH, and CH intersect the circumcircle of traingle ABC at A prime, B prime and C prime. We know angle AHB : angle BHC : angle CHA = 2 : 5 : 8. Find angle AprimeBprimeCprime in degrees.
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Question 1210565: Points $L$ and $M$ lie on a circle $\omega_1$ centered at $O$. The circle $\omega_2$ passing through points $O,$ $L,$ and $M$ is drawn. If the measure of arc $PQ$ in circle $\omega_1$ is $40^\circ,$ then find the measure of arc $LM$ in circle $\omega_1$, in degrees.
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Question 1210566: Trapezoid $HGFE$ is inscribed in a circle, with $\overline{EF} \parallel \overline{GH}$. If arc $EG$ is $40$ degrees, arc $EH$ is $120$ degrees, and arc $FG$ is $20$ degrees, find arc $EF$.
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Question 1210567: In cyclic quadrilateral $PQRS,$
\angle P = 30, \angle Q = 60, PQ = 4, QR = 8.
Find the largest side in quadrilateral $PQRS,$ in degrees.
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Question 1210564: In the figure, if the measure of arc $FG$ is $118^\circ$, the measure of arc $FQ$ is $12^\circ$, and $FR = GR,$ then what is $\angle GRP$, in degrees?
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Question 1210569: In rectangle $EFGH$, let $M$ be the midpoint of $\overline{EF}$, and let $X$ be a point such that $MH = MX$, as shown below. If $\angle EMH = 19^\circ$ and $\angle MEG = 44^\circ,$ then find $\angle GEH,$ in degrees.
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Question 1210568: In the diagram, chords $\overline{XY}$ and $\overline{VW}$ are extended to meet at $U.$ If $\angle UXY = 25^\circ$, minor arc $VW$ is $155^\circ$, and minor arc $XY$ is $82^\circ$, find arc $UW$, in degrees.
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Question 1210570: Let P_1 P_2 P_3 \dotsb P_{10} be a regular polygon inscribed in a circle with radius $1.$ Compute
P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1
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Question 1210572: Compute \angle AGD, in degrees.
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Question 1210571: Find the radius of the quarter-circle.
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Question 1210555: In the diagram below, chords $\overline{AB}$ and $\overline{CD}$ are perpendicular, and meet at $X.$ Find the diameter of the circle.
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Question 1210556: Semicircles are drawn on diameters \overline{AB} and \overline{CD}, as shown below. Find AB.
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Question 1210557: The circle below centered at $O$ has a radius of $5.$ Find $CD.$
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Question 1210558: The circle below centered at $O$ has a radius of $5.$ Find $CD.$
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Question 1210559: Two quarter-circles are drawn inside a unit square. A smaller square is inscribed in the two quarter-circles. Find the area of the smaller square.
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Question 1210554:
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Question 1210551: In equiangular octagaon EFGHIJKL, we know that EF = GH = IJ = KL = 1 and FG = HI = JK = LE = sqrt(2). Find the area of the octagon.
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Question 1210551: In equiangular octagaon EFGHIJKL, we know that EF = GH = IJ = KL = 1 and FG = HI = JK = LE = sqrt(2). Find the area of the octagon.
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Question 1210549: In the diagram below, chords $\overline{AB}$ and $\overline{CD}$ are perpendicular, and meet at $X.$ Find the diameter of the circle.
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Question 1210548: In triangle $XYZ,$ circles are drawn centered at $X$, $Y$, and $Z$, so that all pairs of circles are externally tangent. If $XY = 2,$ $XZ = 2,$ and $YZ = 2$, then find the sum of the areas of all three circles.
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Question 1210550: Trapezoid $ABCD$ is inscribed in the semicircle with diameter $\overline{AB}$, as shown below. Find the radius of the semicircle.
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Question 1210541: In quadrilateral ABCD, the longest side is AB, and the shortest side is CD. Which of the following inequalities must hold? Select all that apply.
AB + AC > BD + BC
AB > (BD + AC)/2
BC > (AB + AC)/2
BD > AC
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Question 1210540: In triangle ABC, let D be a point on side BC. Select all the true statements.

If AD is an altitude of triangle ABC, then AC > AD.
If AD is a median of triangle ABC, then BD > CD.
If AD is an angle bisector of triangle ABC, then AB > BD.
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Question 1210542: All the sides of a triangle have integer length. The perimeter of the triangle is 10, and the triangle is scalene. How many such non-congruent triangles are there?
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Question 1210522: Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.

An equilateral triangle, a regular octagon, a square, a regular pentagon, and a regular n-gon, all with the same side length, also completely surround a point. Find n.
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Question 1210534: Point P lies in regular hexagon ABCDEF such that [ABP] = 3, [CDE] = 5, and [EFA] = 8. Compute [ABC].
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Question 1210535: The diagram shows a square within a regular nonagon. Find $x,$ in degrees.
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Question 1210533: Let PQRST be an equilateral pentagon. If the pentagon is concave, and \angle P = \angle Q = 135^{\circ}, then what is the degree measure of \angle T?
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Question 1210528: ABCDEF is a concave hexagon with exactly one interior angle greater than 180 degrees. (The diagram below is not drawn to scale.) Noah measured the marked angles, getting 90, 120, 45, 75, 150, 180, 15, 60, 30. What interior angle measure of the hexagon, in degrees, is missing from Noah's list?
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Question 1210520: A regular hexagon has a perimeter of p (in length units) and an area of A (in square units). If A=3/2 then find the side length of the hexagon.
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Question 1210530: Grogg draws an equiangular polygon with g sides, and Winnie draws an equiangular polygon with w sides, where g < w. If the exterior angle of Grogg's polygon is congruent to six times the interior angle of Winnie's polygon, find w.
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Question 1210529: Let ABCDE be an equilateral pentagon. If the pentagon is concave, and angle A = angle B = 135, then what is the degree measure of angle E?
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Question 1210531: In the regular octagon below, find x.
https://web2.0calc.com/api/ssl-img-proxy?src=%2Fimg%2Fupload%2Fb8a0eeb81cd61ebb5182a8a7%2Fregular-octagon2.jpg
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Question 1210519: In a certain regular polygon, the measure of each interior angle is 2 times the measure of each exterior angle. Find the number of sides in this regular polygon.
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Question 1210517: Trapezoid $ABCD$ has bases $\overline{AB}$ and $\overline{CD}$. The extensions of the two legs of the trapezoid intersect at $P$. If $[ABC]=3$ and $[PAQ]=8$, then what is $[BDE]$?
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Question 1210516: The perimeter of a rectangle is 40, and the length of one of its diagonals is 10 \sqrt{2}. Find the area of the rectangle.
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Question 1210518: The diagram below consists of a small square, four equilateral triangles, and a large square. Find the area of the large square.
https://web2.0calc.com/api/ssl-img-proxy?src=%2Fimg%2Fupload%2Faf6a1af0cd0ede49901885c6%2Fsquare-triangles.png
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Question 1210526: Let B, A, and D be three consecutive vertices of a regular hexagon. A regular pentagon is constructed on \overline{AB}, with a vertex C next to A. Find \angle BAD, in degrees.
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Question 1210523: The interior angles of a polygon form an arithmetic sequence. The difference between the largest angle and smallest angle is $148^\circ$. If the polygon has 3 sides, then find the smallest angle, in degrees.
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Question 1210524: Let IJKLMN be a hexagon with side lengths IJ = LM = 2, JK = MN = 2, and KL = NI = 2. Also, all the interior angles of the hexagon are equal. Find the area of hexagon IJKLMN.
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Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295, 2296..2340, 2341..2385, 2386..2430, 2431..2475, 2476..2520, 2521..2565, 2566..2610, 2611..2655, 2656..2700, 2701..2745, 2746..2790, 2791..2835, 2836..2880, 2881..2925, 2926..2970, 2971..3015, 3016..3060, 3061..3105, 3106..3150, 3151..3195, 3196..3240, 3241..3285, 3286..3330, 3331..3375, 3376..3420, 3421..3465, 3466..3510, 3511..3555, 3556..3600, 3601..3645, 3646..3690, 3691..3735, 3736..3780, 3781..3825