# High School Practice Problems: Geometry

*Examples from Standards Revision and GLEs
*HG-69) Answer: 2 times that of the original coins

**Two coins, both silver dubloons, are melted down and recast as a single coin of the same thickness. How does the diameter of the new coin compare with the diameter of one of the original coins?**Explain your answer in detail.

HG-68) Answer: intersection is 15 feet above the ground; if 120 feet apart it would still be 15 feet

**Two poles, 60 feet tall and 20 feet tall, stand on opposite sides of a field. The poles are 80 feet apart. Support cables are placed from the top of one pole to the bottom of the opposite pole. How far above the ground is the intersection of the cables? What if the poles were 120 feet apart? **Explain your answer in detail.

HG-67) Answer: 6x6x6

**A cake in the form of a cube falls into a vat of frosting and comes out frosted on all six faces. The cake is then cut into smaller cubes, each one inch on an edge. The cake is cut so that the number of pieces having frosting on three faces will be one-eighth the number of pieces having no frosting at all. There are to be exactly enough pieces of cake for everyone. How many people will receive a piece of cake with frosting on exactly three faces? On exactly two faces? On exactly one face? On no faces? How large was the original cake? **Explain your answer in detail.** **

HG-66) Answer: 8D=44.4 inches, D=5.6 inches

**There's an antique bike parade in town. Stuart has a bike that his great grandfather had given him in which the radius of the front wheel is 8 times the radius of the rear wheel. When the bike travels 100 feet, the number of rotations made by the smaller wheel is 60 more than the number of rotations made by the larger wheel. Find the diameter of each wheel to the nearest tenth of an inch. **Explain your answer in detail.** **

HG-65) Answer: 4 units, 2 units

**The local recycling plant has just bought a new metal compactor that produces a smaller cube of scrap iron than does the older machine. Somebody noticed, however, that the combined volumes of one cube from each compactor was numerically the same as the combined lengths of all their edges. What are the dimensions of the cubes, if you consider only integral solutions?** Explain your answer in detail.

HG-64) Answer: There are actually 3 different parallelograms with the given 3 vertices. The missing vertex not only could have been at (1,8) but also at (-3,0) or at (5,2)

**The ****Lenape**** ****Valley**** ****High School**** Math Club was out on its field day. The teacher, Mr. Romeo, assigned Pam and Jason the problem of finding a buried box. He told them that the box was buried at the fourth vertex of the parallelogram having three of its vertices at (-1,4), (1,1), and (3,5) on the Cartesian grid that was laid out on the field. Pam and Jason dug at (1,8), but the box was not there. Why didn't they find the buried box? What would you do? Defend your decision. **Explain your answer in detail.

HG-63) Answer: 24

**Below is a 4x4x4 cubic block of wood. Suppose all six faces of the cube are painted red and the cube is then cut into 1x1x1 cubes along the lines shown. How many 1x1x1 cubes will have red paint on just two faces? ****? **Explain your answer in detail.** **

HG-62) Answer: 62

**In the figure below, all comer angles are right angles and each number represents the unit-length of the segment which is nearest to it. How many square units of area does the figure have? **Explain your answer in detail.** **

HG-61) Answer: 48 units

**The figure below is divided into 8 congruent squares as shown. The area of the figure is 72 square feet. What is the length of the darkened border in feet? **Explain your answer in detail.** **

HG-60) Answer: 90

**Three squares each have sides of length 6 units and overlap each other as shown below. The points where the sides cross are midpoints. Find the area of the shaded figure in square units. **Explain your answer in detail.** **

HG-59) Answer: 35

**The tower below is made up of five horizontal layers of cubes with no gaps. How many individual cubes are in the tower? **Explain your answer in detail.

HG-58) Answer: 104

**A rectangular picture frame is 12 inches wide and 18 inches high. This includes a 2-inch border (shaded region) around the picture itself. How many square inches are in the border? **Explain your answer in detail.

HG-57) Answer: 50

**Square ABCD has all four of its vertices on a circle with diameter 10 units in length. Segments AC and BD are diameters. How many square units of area does square ABCD have?** Explain your answer in detail.

HG-56) Answer: 5

**The perimeter of a rectangle is 20 feet and the foot-measure of each side is a whole number. How many rectangles with different shapes satisfy these conditions? **Explain your answer in detail.** **

HG-55) Answer: 440

**The front wheel of a vehicle has a circumference of 3 feet, the rear wheel has a circumference of 4 feet. How many more complete turns will the front wheel make than the rear wheel in traveling a distance of I mile on a straight road? (I mile = 5280 feet) **Explain your answer in detail.** **

HG-54) Answer: 9

**ABCD is a rectangle with area equal to 36 square units. Points E, F, and G are midpoints of the sides on which they are located. How many square units are there in the area of triangle EFG? **Explain your answer in detail.** **

HG-53) Answer: 5

**The perimeter of a rectangle is 22 inches and the inch-measure of each side is a natural number. How many different areas in square inches can the rectangle have? **Explain your answer in detail.** **

HG-52) Answer: 38

**Each of the boxes in the figure at the right is a square. How many different squares can be traced using the lines in the figure? **Explain your answer in detail.

HG-51) Answer: 11

**In the rectangle below, line segment MN separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when 4 line segments are drawn through the rectangle? **Explain your answer in detail.** **

HG-50) Answer: 144

**Rectangular cards, 2 inches by 3 inches, are cut from a rectangular sheet 2 feet by 3 feet. What is the greatest number of cards that can be cut from the sheet?** Explain your answer in detail.

HG-49) Answer: 8

**Below is a 3 by 3 by 3 cube. Not all of the cubes are visible. Suppose the entire outside of the cube is painted red including the bottom. How many different 2 by 2 by 2 cubes- with exactly three red faces can be found in the shown cube? **Explain your answer in detail.

HG-48) Answer: 30 meters

**Square ABCD and rectangle AF-FG each have an area of 36 square meters. E is the midpoint of AB. What is the perimeter of rectangle AEFG? **

HG-47) Answer: 36

**Twenty-four meters of fencing is used to fence a rectangular garden.: Let M represents the number of square meters in the area of the garden. What is the largest value that M could have? **Explain your answer in detail.** **

HG-46) Answer: 78

**The “staircase” below is 4 units tall and contains 10 unit squares. Suppose the staircase were extended until it was 12 units tall. How many unit squares would it then contain all together? **Explain your answer in detail.** **

HG-45) Answer: 246

**A rectangular garden is 14 ft. by 21 ft. and is bordered by a concrete walk 3 ft. wide as shown. How many square feet are in the surface area of just the concrete walk? **Explain your answer in detail.** **

HG-44) Answer: 10

**ABCD is a rectangle whose sides are 3 units and 2 units long. The length of the shortest path from A to C following the lines of the diagram is 5 units. How many different shortest paths are there from A to C? **Explain your answer in detail.** **

HG-43) Answer: 150 square units

**Rectangle ABCD contains 3 small congruent rectangles. If the smaller dimension of one of the small rectangles is 5 units, what is the area of rectangle ABCD in square units? **Explain your answer in detail.** **

HG-42) Answer: 9 square units

**Triangle ABC has its vertices A, B, and C on the sides of a rectangle 4 units by 5 units as shown. What is the area of triangle ABC in square units? **Explain your answer in detail.** **

HG-41 Answer: 6

**The length of the shortest trip from A to B along the edges of the cube shown is the length of 3 edges. How many different 3-edge trips are there from A to B? **Explain your answer in detail.** **

HG-40) Answer: 6 square meters

**ABCD is a square with area 16 sq. meters. E and F are midpoints of sides AB and BC, respectively. What is the area of trapezoid AEFC, the shaded region? **Explain your answer in detail.** **

HG-39) Answer: 100 square centimeters

**As shown, ABCD and AFED are squares with a common side AD of length IO cm. Arc BD and arc DF are quarter-circles. How many square cm. are in the area of the shaded region? **Explain your answer in detail.** **

HG-38) Answer: 20

**The six faces of a three-inch wooden cube are each painted red. The cube is then cut into one-inch cubes along the lines shown in the diagram. How many of the one-inch cubes have red paint on at least two faces? **Explain your answer in detail.** **

HG-37) Answer: 38

**The structure below is made of unit cubes piled on top of each other. Some cubes are not visible. What is the number of cubes in the structure? **Explain your answer in detail.

HG-36) Answer: 32

**ABCD is a square with diagonal AC 8 units long. How many square units are in the area of the square? **Explain your answer in detail.** **

HG-35) Answer: 720

**A tractor wheel is 88 inches in circumference. How many complete turns will the wheel make in rolling one mile on the ground? (1 mile = 5,280 feet) **Explain your answer in detail.** **

HG-34) Answer: 8

**A wooden block is 4 inches long, 4 inches wide, and I inch high. The block is painted red on all six sides and then cut into sixteen I inch cubes. How many of the cubes each have a total number of red faces that is an even number? ** Explain your answer in detail.

HG-33) Answer: 4

**Eight one-inch cubes are put together to fomi the T-figure shown at the right. The complete outside of the T-figure is painted red and then separated into one-inch cubes. How many of the cubes have exactly four red faces? **Explain your answer in detail.** **

HG-32) Answer: E, A, N in that order

**Below are three views of the same cube. What letter is on the face opposite (1) H, (2) X, and (3) Y? (Give your answer in the same order.) **Explain your answer in detail.** **

HG-31) Answer: 8

**The complete outside including the bottom of a wooden 4 inch cube is painted red. The painted cube is then cut into 1 inch cubes. How many of the 1 inch cubes do not have red paint on any face? **Explain your answer in detail.** **

HG-30) Answer: 6

**Below, ABCD is a square whose sides are each 2 units long. The length of the shortest path from A to C following the lines of the diagram is 4 units. How many different shortest paths are there from A to C? **Explain your answer in detail.** **

HG-29) Answer: 22

**Each of the boxes in the figure below is a square, Using the lines of the figure, how many different squares can be traced? **Explain your answer in detail.** **

HG-28) Answer: 10

**An acute angle is an angle whose measure is between 0° and 90°. Using the rays in the diagram, how many different acute angles can be formed? **Explain your answer in detail.** **

HG-27) Answer: 16

**The figure shown consists of 3 layers of cubes with no gaps. Suppose the complete exterior of the figure (including the bottom) is painted red and then separated into individual cubes. How many of these cubes have exactly 3 red faces? **Explain your answer in detail.** **

HG-26) Answer: 20

**A square has an area of 144 square inches. Suppose the square is partitioned into six congruent rectangles below. How many inches are there in the perimeter of one of the six rectangles? **Explain your answer in detail.** **

HG-25) Answer: 48

**The set of stairs shown below is constructed by placing layers of cubes on top of each other. What is the total number of cubes contained in the staircase? **Explain your answer in detail.** **

HG-24) Answer: 24 meters

**A square is divided into three congruent rectangles as shown below. Each of the three rectangles has a perimeter of 16 meters. How many meters are in the perimeter of the square? **Explain your answer in detail.** **

HG-23) Answer: 24

**A square piece of paper is folded in half as shown and then cut into two rectangles along the fold. The perimeter of each of the two rectangles is 18 inches. What is the perimeter of the original square? **Explain your answer in detail.** **

HG-22) Answer: #1 (almost half a meter)

**Imagine a wire band fitted snugly around the earth at the equator. If you cut the band, add a piece exactly 10 feet long, reform the band, and then hold it in a position concentric to the equator, which of the following best describes the space between the earth's surface and the wire band? 1) You can just crawl under the band. 2) You can just slide a piece of paper under the band. 3) You can walk upright under the band. **Explain your answer in detail.** **

HG-21) Answer: 96

**A rectangular tile is 2 inches by 3 inches. What is the least number of tiles that are needed to completely cover a square region 2 feet on each side? **Explain your answer in detail.** **

HG-20) Answer: 40

**The square at the right is divided into four congruent rectangles. The Perimeter of each of the four congruent rectangles is 25 units. How many units are there in the Perimeter of the square? **Explain your answer in detail.

HG-19) Answer: 58

**In the figure below, each number represents the length of the segment which is nearest it. How many square units are in the area of the figure if there is a right angle at each comer of the figure? **Explain your answer in detail

HG-18) Answer: 36

**Each of the small boxes in the figure below is a square. The perimeter of square ABCD is 36 cm. What is the perimeter of the figure shown with darkened outline? **Explain your answer in detail.

HG-17) Answer: 70

**The tower shown below is made of horizontal layers of unit cubes, not all being visible in the diagram. How many unit cubes are contained in the tower? **

HG-16) Answer: 11

**The tower below has no gaps. Suppose it is painted red on all exterior sides including the bottom,, and then cut into cubes along the indicated lines. How many cubes will each have red paint on just three faces? **Explain your answer in detail.** **

HG-15) Answer: 40

**Each of the small boxes in the figure below is a square and the area of the figure is 52 square units. How many units are there in the outer perimeter of the figure? **Explain your answer in detail.** **

HG-14) Answer: 395,840,674.4 sq. inches

**The Acme Tinplate Corporation has been asked to manufacture one million cans for a new product about to go to market. Each can is to be cylindrical, with a radius of 3 inches and a height of 6 inches. How many square feet of tinplate are used in making the cans? **Explain your answer in detail.** **

HG-13) Answer: 8

**The length of AE is 20 cm. B is the midpoint of AC, C is the midpoint of BD, and D is the midpoint of BE. What is the length of DE in cm? **Explain your answer in detail.** **

HG-12) Answer: 4 square units

**Below, there are two large congruent squares with sides 7 units long and three small congruent squares with sides 3 units long. If the shaded figure is also a square, what is its area in square units? **Explain your answer in detail.** **

HG-11) Answer: 4 1/2 inches

**An automobile driveway is 54 feet long by 8 feet wide. It is to be covered with blacktop. If 3 loads of blacktop are available for the job, how deep a layer of blacktop is possible? (One load =2 cubic yards.) **Explain your answer in detail.

HG-10) Answer: 20

**The tower shown below is made of unit cubes stacked on top of each other. Some of the unit cubes are not visible. How many unit cubes are not visible? **Explain your answer in detail

HG-9) Answer: 11

**What is the greatest number of points of intersection that can occur when 2 different circles and 2 different straight lines are drawn on the same piece of paper? **Explain your answer in detail.** **

HG-8) Answer: 1/8

**ABCD is a square, and E and F are midpoints of sides AD and AB respectively as shown. What fractional part of the total area of the square is the area of triangle AEF? **Explain your answer in detail.** **

HG-7) Answer: 30

**Each of the small boxes in the diagram below is a square and congruent to each of the others. How many different squares can be traced using the lines of the diagram as sides? **Explain your answer in detail.** **

HG-6) Answer: 27

**The tower shown below is made by placing congruent cubes on top of each other. Not all cubes of the tower are visible. How many cubes does the tower contain? **Explain your answer in detail.** **

HG-5) Answer: 18

**The cube below is constructed of congruent boards, each being of the same size and shape. How many boards does the cube contain? **Explain your answer in detail.** **

HG-4) Answer: 64

**ABCD is a square with each side divided into three segments of length 1 unit, 8 units, and 1 unit respectively, as shown in the diagram below. What is the sum of the areas of the four shaded triangles? **Explain your answer in detail.**
**

HG-3) Answer: 1/4

**ABCD is a square; E, F, G, and H are midpoints of AP, BP, CP, and DP respectively. What fractional part of the area of square ABCD is the area of square EFGH? **Explain your answer in detail.** **

HG-2) Answer: 12

**ABCD is a square which contains nine small congruent squares as shown. The area of square ABCD is 36 square units. What is the area of triangle ACE in square units? **Explain your answer in detail.** **

HG-1) Answer: 18

**Different rectangles can be traced using the lines in the figure given at the right. How many different rectangles can be traced? **Explain your answer in detail.

Use inductive reasoning to formulate conjectures about geometric relationships based on explorations and experiments. Example: - Investigate the relationship among the medians of a triangle using paper folding or dynamic geometric software.
Use deductive reasoning to prove that a valid mathematical statement is true using paragraph, two-column, or flow-chart formats. Identify errors or gaps in flawed mathematical reasoning, and develop counterexamples to refute invalid statements. Determine the validity of the converse, inverse, and contrapositive of a valid proposition. Examples: - If m and n are odd integers, the sum m + n is an even integer. State the converse and determine whether or not it is valid.
- If a quadrilateral is a rectangle, the diagonals have the same length. State the contrapositive and determine whether or not it is valid.
Describe the structure of an axiomatic system—including the role of undefined terms, definitions, postulates/axioms, and theorems—using examples from geometry. Identify and use necessary and sufficient conditions to define two- and three-dimensional figures. This would include tasks such as describing and classifying quadrilaterals: squares, rectangles, rhombi, parallelograms, trapezoids, and kites. Prove theorems about angles, including angles that arise from parallel lines cut by a transversal. Identify necessary and sufficient conditions to establish congruence and similarity in triangles and use these conditions, along with theorems about angles, to prove conjectures about triangle congruence, triangle similarity, and other properties of triangles. Examples: - Prove that the base angles of an isosceles triangle are congruent.
- Prove that congruent triangles are similar.
- For a given triangle RST, prove that triangle XYZ, formed by joining the midpoints of the sides of triangle RST, is similar to triangle RST.
Define and identify special lines, segments, and rays associated with a triangle; analyze proofs about properties associated with these special lines, segments, and rays; and solve related problems. Lines, segments, and rays would include perpendicular bisectors of sides, medians, angle bisectors, and altitudes; Example: Make and prove conjectures about properties of quadrilaterals and other polygons. This includes symmetries, properties of angles, diagonals, and angle sums. Examples: - Determine the measure of interior and exterior angles of regular polygons and write rules that describe the relationship between the number of sides and the angle measures.
- Prove the opposite sides of a parallelogram are congruent.
Prove and apply the Pythagorean theorem and its converse. Examples: - In rABC, with right angle at C, draw the altitude. Name all similar triangles in the diagram. Use these similar triangles to prove the Pythagorean theorem.
- On each side of a right triangle draw a square with that side of the triangle as a side of the square. Find the area of the three squares. What is the relationship of the areas?
- Apply the Pythagorean theorem to derive the distance formula.
Define and use the basic trigonometric ratios of sine, cosine, and tangent to solve problems. College Readiness Example: - Use the Pythagorean theorem to establish that sin2 θ + cos2 θ = 1 for θ between 0° and 90°.
Use the properties of special right triangles (30°-60°-90° and 45°-45°-90°) to solve problems. Example: - Determine the length of the altitude of an equilateral triangle whose side lengths are equal to 5 units.
Prove and apply properties of angles, arcs, chords, secants, and tangents of circles. Examples: - Prove that a triangle inscribed on the diameter of a circle is a right triangle.
- Prove that if a radius of a circle is perpendicular to a chord of a circle, then the radius bisects the chord.
Perform straightedge-and-compass, paper-folding, or dynamic software constructions to make and test conjectures about geometric properties and relationships and justify the steps in a construction using the theorems and postulates of Euclidean geometry. Examples: - Construct the perpendicular bisector of a line segment. Give the postulate or theorem underlying each step.
- Construct a circle through three non-collinear points. Give the postulate or theorem underlying each step.
- Given a line, and a point not on the line, construct a line through the point and parallel to the given line. Give the postulate or theorem underlying each step.
Describe the intersections of lines on the plane and in space, of a line and a plane, and of two planes in space. Describe regular and non-regular polyhedra using their faces, edges, vertices, and properties. “Describing” might include: - characterizing basic polyhedra, such as prisms, pyramids, parallelepipeds, and tetrahedra;
- specifying the number and shape of the faces;
- specifying properties about the faces or edges;
- specifying the number of vertices and edges; or
- analyzing polyhedra by sketching a corresponding net.
Analyze and describe cross sections formed by the intersection of planes and three-dimensional figures. Example: - Describe all the possible cross sections of a cube cut by a plane not parallel to a face.
Determine the equation of a line on a coordinate plane that is described geometrically, including a line through two given points, a line through a point and a parallel to a given line, and a line through a point and a perpendicular to a given line. Examples: - Write an equation for the perpendicular bisector of a given line segment.
- Determine the equation of a line through the points (5, 3) and (5, -2).
Determine the coordinates of a point meeting certain conditions and the distance between points described geometrically on a coordinate plane. Examples: - Determine the coordinates for the midpoint of a given line segment.
- Given the coordinates of three vertices of a parallelogram, determine all possible coordinates for the fourth vertex.
Find the equation of a circle that is described geometrically on a coordinate plane; given equations for a circle and a line, determine the coordinates of their intersection(s). Examples: - Write an equation for the circle determined by a given center and a radius.
- Write an equation for a circle given a line segment as a diameter.
- Write an equation for a circle determined by a given center and tangent line.
Additional College Readiness Example: - Given the circle x2 + y2 = 4 and the line y = x, find the points of intersection.
Use coordinate geometry to establish and verify properties, prove conjectures, and solve problems about polygons and circles. Examples: - Given four points on a coordinate plane, determine whether they are the vertices of a rhombus.
- Given a parallelogram on a coordinate plane, verify that the diagonals bisect each other.
- Given a chord of a circle, verify that the radius perpendicular to a chord is the bisector of the chord.
Sketch translations, reflections, rotations, dilations, and compositions of these transformations for a given two-dimensional figure on a coordinate plane and make generalizations regarding translations and reflections about the coordinate axes and the line y = x. Make and test conjectures about transformations, their compositions, and their properties. Conjectures might involve compositions of transformations and inverses of transformations, the commutativity and associativity of transformations, and the congruence and similarity of two-dimensional figures under various transformations. Examples: - Identify transformations (alone or in composition) that preserve congruence.
- Determine whether the composition of two line reflections is commutative.
- Determine whether the composition of two rotations about the same point of rotation is commutative.
- Find a rotation that is equivalent to the composition of two reflections over intersecting lines.
- Find the inverse of a given transformation.
Given two congruent or similar figures on a coordinate plane, describe a composition of translations, reflections, rotations, and dilations that superimposes one figure on the other. Describe the symmetries of two- and three-dimensional figures, including reflections across a line and rotations about a point for two-dimensional figures, and reflections across a plane and rotations about a line for three-dimensional figures. |

*Examples of Geometric Sense from the 2006 GLEs – High School*

Make and test conjectures about 2 dimensional and 3 dimensional shapes and their individual attributes and relationships using physical, symbolic, and technological models. Use the relationship between similar figures to determine the scale factor. Match or draw a 3 dimensional figure that could be formed by folding a given net. Match or draw 3 dimensional objects from different views using the same properties and relationships. Sort, classify, and label prisms, cylinders, cones, and pyramids. Sort, classify, and label 2 dimensional and 3 dimensional shapes according to characteristics including faces, edges, and vertices, using actual and virtual modeling. Construct geometric figures, including angle bisectors, perpendicular bisectors, and triangles given specific characteristic, using a variety of tools and technologies. Given a set of characteristics, draw a plane figure and justifies the drawing. Create a three dimensional scale drawing with particular geometric properties. Use properties of triangles and special right triangles in situations. Determine geometric properties of two dimensional objects using coordinates on a grid. Determine the location of a set of points that satisfy given conditions. Represent real life situations on a coordinate grid or describes the location of a point that satisfies given conditions. Use tools and technology to draw objects on a coordinate grid based on given properties. Write ordered pairs to describe the locations of points or objects on a coordinate grid. Use multiple translations, reflections, and/or rotations to create congruent figures on a coordinate grid. Use dilation of a given figure to form a similar figure. Determine the final coordinates of a point after multiple transformations. Describe a combination of two translations, reflections, and/or rotations to transform one figure to another figure with or without a coordinate grid. Determine rotational symmetry of a figure. Use technology to create 2 and 3 dimensional animations using combinations of transformations. |