Geometry: Geometric Operations

How do points transform during a 180 degree rotation

( x , y ) --> ( -x , -y )

if point is roated 90 degrees clockwise, what happens to the points (x,y)?

(x,y) --> (y,-x)

Give an example of special right triangles

30-60-90 right triangles whose acute angles are 30 degrees and 60 degrees

what is the ratio of the surface area of a cone if it is tripled ratio --> old:new

3^2 = 9 1:9

how do you find the area of a trapezoid?

A = 1/2 h (b1 + b2)

What is a rigid transformation?

A geometric transformation after which the image and pre-image are congruent

What is a rotation/turn?

A geometric transformation consisting of a turn of a shape about a point, often the origin, (0,0)

What is a translation/slide?

A geometric transformation consisting of movement to the right or left, up or down, or a combination of movements

What is a glide reflection?

A geometric transformation consisting of two transformations at once: a translation and a reflection

What is a reflection/flip?

A geometric transformation over a line that produces a mirror image of the original object or image

What is a reflection/flip?

A geometric transformation over a line that that produces a mirror image of the original object or image

What is a proof?

A new true statement using multiple axioms and theorems

What is a tessellation?

A pattern of shaped that fit perfectly together

What is a straightedge?

A ruler without markings, which is used to draw straight lines or check for straightness

What is a theorem?

A statement that is accepted because it has been proven

What is a compass?

A tool used to draw circles or arcs

What is an axiom/postulate?

A truth that is accepted as being self-evident, without proof

A pole broke 15 feet from its base. The base remains perpendicular to the ground. The break was not complete, however, so the top portion of the pole extends to the ground while still remaining attached to its base, as shown in the diagram. The top portion of the pole hits the ground 8 feet from the place where the base of the pole enters the ground. How tall, in feet, was the pole before it broke? A. 32 B. 25 C. 23 D. 17

A. 32 a^2 + b^2 = c^2 (15)^2 + (8)^2 = c^2 225 + 64 = c^2 289 = c^2 17 = c^2 15 + 17 = 32

A tree casts a shadow that is 8 yards in length. If the angle between the ground and the top of the tree is 30°, which of the following is closest to the height of the tree? A. 4.6 yds B. 5.7 yds C. 11.3 yds D. 13.9 yds

A. 4.6 yds tan (30) = x/8 8 tan (30) = x

Jim wants to walk to Bill's house. To get to Bill's house, Jim walks 3 miles south and then walks 4 miles east. Jim wants to know how many miles he would have walked if he just walked in a straight line. How many miles would Jim have walked if he walked in a straight line to Bill's house? A. 5 miles B. 6 miles C. 4 miles D. 7 miles

A. 5 miles (3)^2 + (4)^2 = c^2

In triangle XYZ, XY has length 5 and YZ has length 9. For which of the following possible lengths of XZ is XYZ an acute triangle? Select all answers that apply. A. 8 B. 9 C. 10 D. 11

A. 8 B. 9 C. 10 a^2 + b^2 > c^2 (5)^2 + (9^2) > c^2 106 > c^2 10.29 > c

A triangle has side lengths of 5, 8 and x. Which of the following could be the value of x? Select all answers that apply. A. 8 B. 13 C. 12 D. 3

A. 8 C. 12 5 + 8 = 13 8 - 5 = 3

Mr. Stevens is teaching his class to use trigonometry rules to solve for unknown side lengths and angles. One student starts using SOH CAH TOA on all triangles in class and is frustrated when he gets the incorrect result. How should Mr. Stevens help the student? A. Explain that SOH CAH TOA is only for triangles that contain a right angle. B. Teach the student law of cosines because that can be used on all triangles. C. Teach the student the Pythagorean theorem because then he can solve every triangle problem. D. Teach the student law of sines because that can be used on all triangles.

A. Explain that SOH CAH TOA is only for triangles that contain a right angle.

Ms. Klein is teaching her students about tessellations. She brings in magnetic tiles for her students to create their own tessellations as an introductory activity. She hands them out to the students and then begins to explain the activity for the day. Students are not paying attention and instead building whatever they want. How can she improve her teaching practice? A. Give the students clear instructions and a worksheet that accompanies the activity prior to handing out the tiles. B. Take away tiles from the misbehaving students. C. Give students a picture of a tessellation to color instead. D. Do not use magnetic tiles because they are distracting to the students.

A. Give the students clear instructions and a worksheet that accompanies the activity prior to handing out the tiles.

If the lengths of two sides of a triangle are given as 5 cm and 11 cm, what statement is true of the third side? A. The third side is greater than 6 cm but less than 16 cm. B. The third side is greater than 5 cm but less than 16 cm. C. The third side is greater than 5 cm but less than 11 cm. D. The third side is greater than 6 cm but less than 11 cm.

A. The third side is greater than 6 cm but less than 16 cm. 11+5 = 16 11-5=6

A cube has sides of 2 cm each. What happens to the volume of the cube if the length of each side is tripled? A. The volume becomes 27 times larger. B. The volume becomes 3 times larger. C. The volume becomes 6 times larger. D. The volume becomes 8 times larger.

A. The volume becomes 27 times larger. Cube 1: 2 x 2 x 2 = 8 Cube 2: 6 x 6 x 6 = 216 216 / 8 = 27 times larger

What tool is used to make sure a line is straight? A. straightedge B. protractor C. compass D. scale

A. straightedge

formula for area of a triangle

A=½bh

A triangle has coordinate points A (-2, -1), B (-6, -1), and C (-4, 5). If it is rotated 90° clockwise about point A, what are the new coordinates of point C? A. (-4, -5) B. (4, 1) C. (-2, 3) D. (-2, -1)

B. (4, 1) The base of the triangle is 4 units long and the height is 6 units tall. Point B will be 4 units away from point A and because of the rotation, 4 is added to the y-value of A. The x-value will be the same as A. Point B is (-2, 3). Point C is centered above the midpoint of line AB. The midpoint is (-2, 1) and because the triangle is now on its side, the x-value adds 6 and the y-value remains. The new coordinates are (4, 1).

A triangle with coordinates (2, -4), (3, 2), and (0, 0) is moved 3 units right and 3 units down. What are the coordinates of the new image? A. (5, 7), (6, 5), (3, -3) B. (5, -7), (6, -1), (3, -3) C. (5, -7), (6, 5), (3, 3) D. (5, -7), (1, -7), (3, -3)

B. (5, -7), (6, -1), (3, -3)

The right circular cone shown has a height of 30 units and a base radius of 6 units. The cone is going to be truncated 10 units from the top. What fraction of the volume of the entire cone will the smaller (shaded) cone hold? (The volume, V, of a right circular cone with base radius r and height h is given by the formula V = (1/3)​πr^2h A. 1/9 B. 1/27 C. 1/6 D. 1/3

B. 1/27

The points X, Y, Z lie in a plane. If XY = 16.3 and XZ = 9.8, which of the following numbers could be the length of YZ? A. 27 B. 19 C. 6 D. 32

B. 19 16.3 + 9.8 = 26.1 16.3 - 9.8 = 6.5

One leg of a right triangle is 17 inches less than the other leg. How long does the shorter leg need to be in order for the hypotenuse to be at least 25 inches? A. -7 inches B. 7 inches C. 17 inches D. 24 inches

B. 7 inches x^2 + (x-17)^2 = (25)^2

For triangle ABC where BC is the hypotenuse, if AB=5, BC=10, which of the following is not a possible measurement for AC for the triangle to be obtuse? A. 6 B. 9 C. 7 D. 8

B. 9 a^2 + b^2 < c^2 (5)^2 + b^2 < (10)^2 25 + b^2 < 100 b^2 < 75

Mr. Owens wants encourage his students to write about math. As a homework assignment, He asks his students to write about why it is beneficial to take the diagonal across the park using mathematical reasoning. Some of the responses are only one sentence long. How can he improve this activity in the future? Select all answers that apply. A. Create a secondary multiple choice exam relating to the topic. B. Give students a sample response to a prompt about a prior topic that exemplifies properties of good writing. C. Model writing responses to word questions in class. D. Give students a rubric for how the assignment will be graded.

B. Give students a sample response to a prompt about a prior topic that exemplifies properties of good writing. C. Model writing responses to word questions in class. D. Give students a rubric for how the assignment will be graded.

A teacher wants to introduce triangle inequality to his students. Which of the following is a best first activity to introduce the concept? A. Give students a quiz relating to possible side length combinations of triangles. B. Give students sticks of various lengths and ask them to make as many triangles as possible while recording combinations that work and don't work. C. Give students a worksheet which gives formulas relating to triangle inequality. D. Give students paper and ask them to draw triangles of different shapes.

B. Give students sticks of various lengths and ask them to make as many triangles as possible while recording combinations that work and don't work.

Given the following glide reflection parameters, what are the coordinates of vertex C′′? The vertices of △ABC are A(2,3), B(3,6), and C(7,1) Translation: (x,y)→(x−11,y) Reflection: in the x-axis. A. ( −4 , 1 ) B. ( 4 , 1 ) C. ( −4 , −1 ) D. ( 4 , -1 )

C. ( −4 , −1 )

The side lengths of parallelogram X are 2 and 3. A similar parallelogram Y has side lengths 10 and 15. The area of parallelogram Y is how many times the area of parallelogram X? A. 3 B. 5 C. 25 D. 125

C. 25 2:3 10:15 times 5 in each ratio of areas is squared 5 x 5 = 25

A ladder, 35 feet in length, leans against the side of a building. The base of the ladder is 21 feet from the base of the building. How far above the ground does the ladder touch the building? A. 14 feet B. 17.5 feet C. 28 feet D. 40.8 feet

C. 28 feet a^2 + b^2 = c^2 (21)^2 + b^2 = (35)^2 441 + b^2 = 1225 b^2 = 784

Sam is at a Christmas tree farm to pick out the perfect tree. The tree he likes is not labeled with its height, but Sam wants to make sure it will fit inside his house. If the angle of elevation to the top of the tree is 30 \degree30° and the distance from the angle of elevation to the center of the tree trunk is 1212 ft, which of the following is closest to the height of the tree? A. 10 ft B. 6 ft C. 7 ft D. 9 ft

C. 7 ft tan (30) = x/12 12 tan (30) = x

Mr. Francis has been teaching his students about how dimensional change affects volume and area. He asks students what would happen to the volume of a cube if he doubles the side length. Several students say it will quadruple. How should he address this misunderstanding? A. Ask them what would happen if he triples side length instead. B. Make nets of two different cubes one which has side length double the other and have students assemble the cubes from the nets to make the comparison. C. Give a problem with real numbers and have students show their calculations. D. Tell students that it will increase by a factor of 8.

C. Give a problem with real numbers and have students show their calculations.

Mrs. Blue wants her students to be able to write two column geometric proofs. Which is the most appropriate way to determine their mastery? A. Give students a proof with steps missing and ask them to write in the missing reasons. B. Have students write an essay about writing proofs. C. Give students an open ended exam where they write multiple two column proofs. D. Give students a multiple choice exam related to proofs.

C. Give students an open ended exam where they write multiple two column proofs.

Which of the following cannot form a regular tessellation? A. hexagon B. triangle C. pentagon D. square

C. pentagon

What is a special right triangle?

Common right triangles in which the angle measures and/or side measures have a special relationship

In rectangle ABCD, point A is located at (-3,6) and point D has coordinates (-3, -4). If ABCD is rotated 90° clockwise about point D, what are the coordinates of A'? A. (10, 0) B. (3, -6) C. (6, -3) D. (7, -4)

D. (7, -4) D: (-3,-4) D: (0,0) + 3 + 4 A: (-3,6) A: (0,10) (x,y) --> (y,-x) (10,0) - 3 - 4 (7,-4)

A trapezoid has an area of 600 mm^2. What is the area of the new trapezoid formed by dilating the original by a factor of 3? A. 200 mm2 B. 500 mm2 C. 1800 mm2 D. 5400 mm2

D. 5400 mm2 3^2 = 9 600(9)

A triangle has side lengths of 7, 14, and x. Which of the following could be the length of x? A. 22 B. 6 C. 24 D. 8

D. 8 7 + 14 = 21 14 - 7 = 7

Maria has drawn a triangle with sides 3" and 8". Which of the following is a true statement? A. The third side could be 4" long. B. The third side could be 12" long. C. The third side must be less than 8" long. D. The third side must be greater than 5" long.

D. The third side must be greater than 5" long. 8-3 = 5 8+3 = 11

What tool should Jenny use to draw a circle? A. ruler B. straightedge C. compass D. protractor

D. protractor

A student is instructed to draw a four-pointed geometric shape on an xy-plane. After the shape is drawn, the student is instructed to add 5 to each x-coordinate and add 3 to each y-coordinate. Which of the following did the student perform? A. rotation B. reflection C. refraction D. translation

D. translation

What is the parallel postulate?

Given a line and a point not on that line, there exists a unique line through the point parallel to the given line Related: if two straight lines are intersected by a line segment and form two angles which sum to less than 180 degrees, then the two lines will intersect on the same side as the angles

Give an example of an axiom/postulate

Given any two distinct points, there is a line that contains them

Give an example of a theorem

If a transversal intersects two parallel lines, then alternate interior angles are congruent

What is the right triangle property: leg-angle congruence?

If one leg and an acute angle of a right triangle are congruent to one leg and the corresponding acute angle of another right triangle, then the triangles are congruent

What is the right triangle property: hypotenuse-leg congruence?

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent

What is the right triangle property: hypotenuse-angle congruence (HA)?

If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and corresponding acute angle to another right triangle, then the triangles are congruent

What is the right triangle property: leg-leg congruence?

If the legs of a right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.

What is the triangle: altitude?

Of a triangle, the line segment from any vertex perpendicular to the opposite side

What is the exterior angle of a polygon?

The angle between a side and an adjacent side extended, on the exterior of a closed shape

What is an interior angle?

The angle formed on the interior of a closed shape

What is a hypotenuse?

The longest side of the right triangle and is always directly opposite the right angle, an angle of 90 degrees

what is a reflection over x=y

The reflection of a point over the line x = y causes the x and y values to swap.

What is the triangle inequality theorem?

The sum of any 2 sides of a triangle must be greater than the measure of the third

What is the pythagorean theorem?

The sum of the squares of the legs

Give an example of rigid transformations

Translations, reflections, rotations, and glide reflections

What is a right triangle?

Triangle with one right angle measure exactly 90 degrees

What is a right triangle?

Triangle with one right angle measuring exactly 90 degrees

what is the reflexive property of equality

a = a

what is the transitive property of equality

a = b b = c therefore a = c

how do you find the possible third length of an obtuse triangle?

a^2 + b^2 < c^2 two lengths must be less than third length

What is the Pythagorean Theorem?

a^2 + b^2 = c^2

what is the formula for perimeter of triangle?

add all sides

How can you find a potential third length given two lengths?

add the two lengths to find out the max possible length subtract the two lengths to find out the minimum possible lengths

what does it mean for something to dilate?

enlarge

In a 45-45-90 triangle what are the side lengths?

side lengths: x hypotenuse: x sqrt 2

How do you find all possible lengths of an acute triangle given two sides

two lengths must be larger than third a^2 + b^2 > c^2