Topics Covered on Geometry Placement Exam


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1 Topics Covered on Geometry Placement Exam  Use segments and congruence  Use midpoint and distance formulas  Measure and classify angles  Describe angle pair relationships  Use parallel lines and transversals  Apply triangle sum properties  Use congruent triangles  Use isosceles and equilateral triangles  Perform congruence transformations  Midsegment theorem  Use perpendicular bisectors  Use angle bisectors of triangles  Use medians and altitudes  Use inequalities in a triangle  Use the hinge theorem  Use similar polygons  Apply the Pythagorean theorem and its converse  Use similar right triangles  Special right triangles  Apply the tangent ratio  Apply the sine and cosine ratio  Solve right triangles  Find angle measures in polygons  Use properties of parallelograms  Use properties of rhombuses, rectangles, squares, trapezoids, and kites  Identify special quadrilaterals  Translate figures and use vectors  Perform reflections, rotations, and dilations  Apply compositions of transformations  Identify symmetry  Use properties of tangents  Find arc measures  Apply properties of chords  Use inscribed angles and polygons  Find segment lengths in circles  Write and graph equations of circles  Find circumference and arc length  Find areas of circles, sectors, and segments  Find areas of regular polygons  Use geometric probability  Find volume of prisms, cylinders, pyramids, cones, and spheres  Find surface area of prisms, cylinders, pyramids, cones, and spheres  Explore similar solids
2 Geometry Placement Exam Review 1) Point B is between A and C on AC. Use the given information to write an equation in terms of x. Solve the equation. Then find AB and BC, and determine whether AB and BC are congruent. a) AB=4 x 5 b) AB=x +3 BC=2 x 7 BC=2 x+1 AC =54 AC =10 2) Find the coordinates of the midpoint of the segment with the given endpoints. A(2, 4), B(7,1) 3) Use the endpoint and midpoint M of the segment to find the coordinates of the other endpoint. A(3, 7),M (1,1)
3 4) Find the length of the segment with given endpoint and midpoint M. A( 3, 4), M (9,5) 5) Use the given information to find the indicated angle measures. 6) 7) 8) c)
4 9) Use the diagram and the given information to solve for each of the angles. a) 1 = b) 2 = c) 3 = d) 4 = e) 5 = f) 6 = g) 7 = h) 9 = i) 8 = j) 10 = k) 11 = 10) a) b) c) d) e) f)
5 11) Find the value(s) of the variables. c) d) 12) Find the values of x and y. c) d)
6 e) 13) Find the measure of the exterior angle. 14)
7 15) Define, write the formula, give the formula, or draw and label a picture to help you remember the following terms, formulas, and/or theorems. a) Midsegment: b) Midpoint Formula: c) Distance Formula: d) Perpendicular Bisector Theorem: e) Converse of Perpendicular Bisector Theorem: f) Angle Bisector Theorem: g) Converse of Angle Bisector Theorem: h) Circumcenter: i) Incenter: j) Concurrency of Perpendicular Bisectors of a Triangle Theorem: k) Concurrency of Angle Bisectors of a Triangle Theorem:
8 16) 17) Find the value of x that makes N the incenter of the triangle. 18) Use the diagram of triangle ABC where D, E, and F are the midpoints of the sides.
9 19) 20) Find the value of x. 21) Suppose that J is the incenter. Show your work. Find the value of x. Find the length of AG. Find the length of JK. 22) Point S is the centroid of triangle PQR. Use the given information to find the value of x.
10 23) A triangle has one side of length 10 and another of length 6. Describe the possible lengths of the third side. 24) Use the Hinge Theorem or its converse and properties of triangles to write and solve an inequality to describe a restriction on the value of x. 25) Find the value of x and y. Then find the following lengths: JK, KM, KL, JM x: 2y 1 4x y 10 y: JK: KM: KL: JM: 26) Find the area and perimeter of the rectangle if AC = 10 and BD = 24 Area: Perimeter:
11 27) Find the area of the triangle, round your answer to three decimal places. 28) Decide whether the numbers can represent the side lengths of a triangle. If they can, classify the triangle as acute, right, or obtuse. a) 26, 35, 62 b) 14, 18, 29 c) 30, 72, 78 d) 17, 19, 22 29) Find the value of the variable.
12 c) d) 30) c) d)
13 31) A symmetrical canyon is 4850 feet deep. A river runs through the canyon at its deepest point. The angle of depression from each side of the canyon to the river is 60 o. Round to the nearest thousandth. a) Find the distance across the canyon. b) Find the length of the canyon wall from the edge to the river. c) Is it more or less than a mile across the canyon? (5280 feet = 1 mile) 32) Find the length of AB. c) 33) Find the value of x. Show your work!
14 c) d) e) f) 34) Find the sum of the measures of the interior angles of the indicated convex polygon. a) Dodecagon b) 24gon 35) Find the value of n for each regular ngon described. a) Each interior angle of the regular ngon has a measure of 162 o. b) Each exterior angle of the regular ngon has a measure of 5 o.
15 36) Find the value of x, and the measure of the missing angles. 37) The side view of a storage shed is shown below. Find the value of x. Then determine the measure of each angle. 38) 39) Find the length of the midsegment of the trapezoid.
16 40) 41) Find the value of x. 42)
17 43) Find the scale factor. Tell whether the dilation is a reduction or an enlargement. Then find the values of the variables. Scale factor: Scale factor: reduction or enlargement: reduction or enlargement: x: x: y: 44) c) 45)
18 46) Find the value(s) of the variable given that P, Q, and R are points of tangency. c) d) 47)
19 48) Find the values of the variables. 49) Find the value of x. c) d) e) f)
20 50) Find the value of x. c) 51) What is the center and radius of a circle with equation (x+5) 2 +( y 2) 2 =100 52) Find the area of the shaded region. Round answers to three decimal places, if necessary.
21 53) Find the length of arc AB. Round answers to three decimal places. 54) Find the probability that a randomly chosen point in the figure lies in the shaded region. 55)
22 56) Find the volume of the right prism. Round your answer to two decimal places, if necessary. 57) Find the volume of the right cylinder. Round your answer to two decimal places, if necessary. 58) Find the volume of the solid. Round your answer to two decimal places, if necessary. 59) Find the radius of a sphere with the given surface area S.
23 60) 61)
24 Geometry Placement Exam Review Answers 1) 1) a) x=11 AB = 39, BC = 15; not congruent b) x=2 AB = 5, BC = 5; congruent 2) ( 9 2, 3 2) 3) ( 1,9) 4) AM = 15, so from endpoint to endpoint = 30 5) 86 o 6) 88 o 7) 17 o 8) a) adjacent b) complementary c) vertical angles & supplementary 9) a) 60 o b) 120 o c) 40 o d) 60 o e) 60 o f) 120 o g) 120 o h) 80 o i) 60 o j) 100 o k) 80 o 10) a) corresponding b) alternate exterior c) consecutive interior d) alternate interior e) corresponding f) alternate interior 11) a) x=53 b) x=40 c) x=110, y=110 d) x=60, y=60 12) a) x=45, y=51 b) x=24, y=66 c) x=22, y=35 d) x=32, y=19 e) x=30, y=13 13) 100 o 14) A ' (0, 1) B ' (3, 0) C ' (2, 5) D ' ( 3, 6)
25 15) a) Midsegment: joins the midpoints of two sides of a triangle such that it is parallel to the 3 rd side of the triangle. b) Midpoint Formula: ( x 1 + x 2 2, y 1 + y 2 2 ) c) Distance Formula: d = (x 2 x 1 ) 2 +( y 2 y 1 ) 2 d) Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. e) Converse of Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. f) Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant fromt eh two sides of the angle. g) Converse of Angle Bisector Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the angle bisector. h) Circumcenter: Point of concurrency of the 3 perpendicular bisectors of a triangle. i) Incenter: Point of concurrency of the 3 angle bisectors of a triangle. j) Concurrency of Perpendicular Bisectors of a Triangle Theorem: The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. k) Concurrency of Angle Bisectors of a Triangle Theorem: The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. l) altitude: The perpendicular segment from a vertex to the opposite side or the line that contains the opposite side. m) median: A segment from a vertex to the midpoint of the opposite side. n) centroid: The point of concurrency of the three medians of a triangle. o) orthocenter: The point of concurrency of the three altitudes of a triangle. p) Concurrency of Medians of a Triangles: The medians of a triangle intersect at a point that is 2/3 the distance from each vertex to the midpoint of the opposite side. q) Concurrency of Altitudes of a Triangles: The lines containing the altitudes of a triangle are concurrent.
26 16) AB = 54 AE = 40 AD = 76 BC = 54 AC = 80 CD = 76 17) x=4 18) 27 19) 51 20) a) x=6 b) x=9 21) x=7 AG = 16 JK = 12 22) x=7 23) 4< x <16 24) x <21 25) x= 1, y=9, JK = 17, KM = 8, KL = 16, JM = 15 26) Area: 60 Perimeter: 34 27) a) b) ) a) no triangle can be formed b) yes, obtuse c) yes, right d) yes, acute 29) a) m=4 b) y=3 c) a=14 d) w=6 30) a) x=18 b) x= c) x=12 d) x=5 31) a) The distance across the canyon is about feet. b) The length of the canyon wall from the edge to the river is about feet. c) It is more than a mile across the canyon. 32) a) AB = 16.2 b) AB = 24 c) AB = 78 33) a) x=20 b) x=42 c) x= 28 3 d) x=8 e) x=10 f) x=27 34) a) 1800 o b) 3960 o 35) a) n = 20 (20gon) b) n = 72 (72gon) 36) x = 4 37) x = 60
27 38) 39) 19 40) 88 o 41) a) x = 14 b) x = ) 88 o 43) a) Scale factor: 5 2 b) Scale factor: enlargement reduction x = 20 x = y = ) c) 45) A 46) a) r= 8 3 b) x=± 2 3 c) x= 1 d) x= ) a) 115 o b) 55 o 48) a) x=25 y=22 b) x=7 y=14 49) a) 55 b) x = 34 c) x = 3 d) x = 21 e) x f) x = 4 50) a) x = 2 b) x = 3 c) x = 14 51) Center: ( 5, 2) Radius: 10 52) a) square units b) square units 53) cm (or 2π ) 54) a) about 47.6% b) about 56.95% 55) 5184 ft 3 56) a) yd 3 b) 210 in 3 57) yd 3 58) a) in 3 b) 126 in 3 59) 1 ft 60) 144 π ft 2 61) The surface area of Pluto is about 1 30 of the Earth's surface area.
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