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GMAT GEOMETRY
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$ Finally, there is one formula involving chords that you may need to know, since questions involving it have appeared a few times in the past In the diagram shown here, the dotted line is a diameter, the solid line is a radius, and the dashed line is a chord When the angle between a chord and a diameter is subtended by an angle between a radius and the same diameter, the measure of the second angle is twice that of the first If that s a mouthful, just study the diagram If you see that, remember that the inside angle is twice the outside angle If you can t remember which is supposed to be larger, then just look to see which appears larger This is probably the one place where the diagram will always be more or less drawn to scale
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VOLUME OF BOXES AND RIGHT CIRCULAR CYLINDERS
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Geometry questions can sometimes become more difficult conceptually when they extend into a third dimension When the GMAT does ask about the volumes of solid figures, it tends to ask about either boxes the three-dimensional forms of squares or rectangles or right circular cylinders, as depicted in the figure Calculating volumes is actually pretty easy: just find the area of the base and multiply that by the figure s height The volume of a cube is the cube of a side, s3 The formula for the volume of a box with a rectangular base is Volume = length width height
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The volume of a right circular cylinder is likewise the area of the base times the height: V = Abh = r 2h since the base of a right circular cylinder is a circle and the area of a circle is r 2
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The area of a triangle: A =
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Heron s Formula for the area of a triangle: A = s(s a )(s b )(s c ), s = 1/2(a + b + c ) 3 s2 The area of an equilateral triangle: A = 4 The Pythagorean Theorem: A 2 + B 2 = C 2 The 3 4 5 triangle: 32 + 4 2 = 52 The 5 12 13 triangle: 52 + 122 = 132 The 30 60 90 triangle: 12 + ( 3 )2 = 22 The area of a parallelogram: Ap = bh The area of an isosceles trapezoid: AT = h(a + x ) or = 1 h(a + b ) 2 The circumference of a circle: c = 2 r The area of a circle: Ao = r 2 The volume of a right circular cylinder: V = r 2h
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1 In a right isosceles triangle, the lengths of the two nonhypotenuse sides are designated a What
is the area of the triangle in terms of a A 1 a 2 3 B 2 a 2 3 C 1 a 2 2 D 3 a 2 2 E 2a 2
2 The US Defense Department has decided that the Pentagon is an obsolete building and that
it must be replaced with an upgraded version: the Hexagon The Secretary of Defense wants a building that is exactly 70 feet high and 200 feet on a side, and that has a hexagonal bulls-eye cutout in the center (somewhat like the current one) that is 50 feet on a side What will be the volume of the new building in cubic feet A 3,937,500 cubic feet B 15,750 cubic feet C 11,550 3 cubic feet D 15,750 3 cubic feet E 3, 937, 500 3 cubic feet
GMAT GEOMETRY
3 What is the area of the shaded figure
A (9 + 3 ) 8 (5 + 3 2 ) B 3 C 9 2 D 3 + 3 2 (9 + 3 ) E 2
4 In the figure above, x = 2 What is the area of circle O
2 B 2 C 2 2 D 4 E 4 2
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5 Which of the following could be the equation of line l in the figure above
A 2x = 3y 2 B 2x = 3y 6 C 3x = 2y 6 D 2 x = 2y 2 3 E 2 2x = 3y 6 3
6 In the rectangular figure above, the area of the shaded region is equal to the area of the non-
shaded rectangle If AB = 6, BC = 4, and the longer side of the non-shaded rectangle is an integer, what is one possible measurement for the shorter side of the non-shaded rectangle A 3 4 3 2
B 1 C
D 12 5 E 16 3
GMAT GEOMETRY
7 In the figure above, what is the value of 2s r
A 60 B 70 C 100 D 140 E 220
The following data sufficiency problems consist of a question and two statements, labeled (1) and (2), in which certain data are given You have to decide whether the data given in the statements are sufficient for answering the question Using the data given in the statements plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you must indicate whether A Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient B Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient C BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient D EACH statement ALONE is sufficient E Statements (1) and (2) TOGETHER are NOT sufficient
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