Geometry Questions

We ve seen that as the sailboat logo is resized by dilation the line segments that make up the logo may be mapped onto parallel lines or stay on the same line The lengths of the image are the lengths of the preimage multiplied by the scale factor Now we will use GeoGebra to compare the angles of a dilated figure to the angles of the original figure Open dilations again Then complete each step below For help watch this video to learn more about measurement tools in GeoGebra Part A Measure and record the measures of these angles in the original logo Then set n 0 5 and n 2 and record the measures of the corresponding angles in each resulting image BIX X 12pt AEVO Angle Original Measure Measure After Dilation n 0 5 FGB 2GBC ZLKJ n 2 cluder formarring 1154 15000

GeoGebra to explore similarity for triangles and similarity transformations Go to similarity for triangles and complete each step below For help watch these short videos about using GeoGebra to measure and transforme figures Part A Find and record the measures of the angles in AABC and ADEF BIV XX 10pt EF Angle Measure Angle Measure ZA ZD 60 ZB ZE 80 20 2F 40 60 BO 40 Space used includes formatting 2047 15000 Hide Sample Answer Angle Measure Angle Measure ZA 4D ZB 20 26 57 135 18 43 ZE 4F 26 57 135 18 43 I Side Length Side Length AB DE Part B Find and record the lengths of the sides in AABC and ADEF Then calculate the ratio of each pair of corresponding sides BIX X 12pt EFE AVE Submitted Pair Ratio DE to AB B

Select the correct answer from each drop down menu reflection 18 16 14 12 10 0 8 6 2 6 R 8 10 12 E F G 14 16 18 PQRS and EFGH are polygons that can be mapped onto each other using similarity transformations The transformation that maps poly PQRS onto polygon EFGH is a rotation followed by dilation with a scale factor of 2 v

Reflectional Symmetry Do the flags below have reflectional symmetry If so draw the appropriate lines of symmetry across each flag Consider how both the shape of the flag the design on the flag would affect this concept 10 stil te

5 How many lines of symmetry do the flags below from various countries have Justify your answer Cuba 9x9 bigh nort thighh wold exostane ort suk Switzerland Jamaica assiq

List 3 different objects that you encounter every day with reflectional symmetry List 3 different objects that you encounter every day with rotational symmetry

Do the items below have rotational symmetry If so by how many compl degrees What s your mathematical reasoning Japan Systemunye 1sbiene0 Ston galt eri od erii peit

6 Identify the transformation below Then describe its scale factor C 9 12 30 15

A man starts walking north at 3 ft s from a point P Five minutes later a woman starts walking south at 4 ft s from a point 500 ft due east of P At what rate are the people moving apart 15 min after the woman starts walking Round your answer to two decimal places ft s

Model a real life staircase using function 18 18 a pi

Question 2 10 points 2 A water slide at an amusement park follows the curve y in m y 2 0 2 0 cos 0 53x 0 40 for 0 x 5 0 Find the angle with the horizontal of the slide for x 2 5 m

Simplify the trigonometric expression cos x sec x tan x

Simplify the trigonometric expression 5 sin t 6 tan t tan t

Simplify the trigonometric expression cos x sin x cos x

Simplify the trigonometric expression 6 sin x sec x tan x

Lesson 21 Practice Problems 1 The triangles here are each obtained by applying rigid motion 5 10 N 11 3 12 8 4 2 The quadrilateral is a parallelogram Find the measure of angles 1 2 and 3 13 9 5 14 10 80 6 15 a Which triangles are translations of triangle 17 Explain how you know 11 b Which triangles are not translations of triangle 1 Explain how you know 7

Cool Down Mystery Quad Quadrilateral ABCD has both reflection and rotation symmetry What type of quadrilateral could ABCD be Show or explain your reasoning

4 Do the shapes below have rotational symmetry Why or why not If so by how many degrees

D Name the transformation that maps the blue pickup truck preimage onto the red pickup image

2 Are the transformations below rigid or non rigid Explain each of your answers in detail o 2

3 How many lines of symmetry exist in each of the flowers below Draw them please

X PY H W BALD Transformations mation given Graph the image of the figure using the transformation given 1 translation 2 units left and 7 units up X nosalamati Name Seda aliny Stmonlean 2 dilation of 2 5 F hodnotbal songs lemey Batkitar T Include Vector Rules X Rule M 2 5x2 5y

23 rotation 90 clockwise about the origin D P X 24 rotation 180 about the origin Z H Y x

se a graphing utility to approximate the local maximum value and local minimum value of the function f x 0 4x 0 6x 3x 4 for 6 x 4 sing a graphing utility graph the function for 6 x 4 and 25 y 10 Choose the correct graph below A Z Q N O B O C F

3 rotation 180 about the origin R Ay B K Z Rule 4 reflection across the y axis ainu noitalansuj S M H F y Rule 2

PRACTICE ANOT Of the infinitely many lines that are tangent to the curve y 8 sin x and pass through the origin there is one that has the largest slope Use Newton s method to find the slope of that line correct to six decimal places

ct the figure as indicated Label the image using prime notation audi ladnd besthet reflection across the x axis tooleretuo 200 poiro 14 reflection across the x axis 10 migno F X Z AX

Rotate the figure as indicated Label the image using prime notation digit 19 rotation 180 about the origin nouselts 20 rotation 90 counterclockwise about the co origin Z H N A D Ay W G

5 W Wy F F Rule x A R R A N N Rule

awise noiloaftur 21 rotation 90 counterclockwise about the origin B N E X 22 rotation 180 about the origin H T y X

17 reflection across the y axis M W N X 18 reflection across the y axis P g

9 translation 5 units left and 2 units up M X 10 translation 2 units right and 5 units down B y Z S X

12 translation 4 units right and 1 unit up Ay A 7 N X

Lesson Cool Down Triangle Angle Sum a Third Way Elena wants to prove that the measures of the 3 angles in a triangle always sum to 180 degrees She starts by using rigid motions to rotate triangle ABC 180 degrees around 2 midpoints 9 c C D B bo B Explain how Elena labeled her diagram and finish her proof MLA m B mcc 180 bo A g C C E

15 reflection across the y axis D P AX 16 reflection across the x axis Toda 081 aolimot Y W y Tapiolandiuto ne X

11 translation 2 units right and 2 units up M U y X 12 translation 4 units right and 1 unit up y A N 1x X

its 7 The 3 circles in the diagram have centers A B and C D B a Explain why segments AB and AC have the same length b Classify triangle ABC Justify your classification c Mai wants to construct a regular hexagon inscribed in the circle centered at C Will these instructions work to finish the hexagon from the construction given If so explain why Mai is correct If not finish the construction correctly i Draw the circle centered at D with radius CD ii Mark the intersection of the circle with the circle centered at C and label that point E iii Draw the circle centered at E with radius DE iv Mark the intersection of the circle with the circle centered at D and labe that point F v Connect ABCDEF to make a regular hexagon

In each question lines AI and GJ are parallel and intersected by the transversal line FE 1 Angles EBI and BCD are corresponding angles Use a transformation that takes angle EBIto angle BCJ to prove that corresponding angles are congruent 1 G C F A 2 Angles ABC and BCJ are alternate interior angles Use a transformation that takes angle ABC to angle BCJ to prove that alternate interior angles are congruent Label any other points on the figure that will help to define a transformation E G B C E B

24 3 Priya followed this set of instructions to make quadrilateral ACBD Choose a description you can be sure is accurate for the shape she constructed 20 2 2p a Draw 2 points A and B b Draw a circle centered at A with radius A B c Draw a circle centered at B with radius AB d Label the intersection points of the circles C and D Draw segments AC BC AD and BD A 2 congruent sides but not all 4 B 4 congruent angles C 4 congruent sides D 4 congruent sides and 4 congruent angles 4 What is the definition of a circle Figura geom trica delimitada por una circu curva cerrada cuyos puntos equidistan a 5 Triangle ABC is isosceles Use straightedge and compass tools to construct the perpendicular bisector fo segment AB r B

21 96 87 125

4 25 40

Brianna is taking part in a lottery for a room in one of two new dormitories at her college She will draw a card randomly to determine which room she will have Each card has the name of a dormitory X avier or Y oung and also a two person room number or an apartment number Forty percent of the available spaces are in X Seventy percent of the available spaces in X are rooms and 20 of the spaces in Y are rooms If Brianna gets an apartment what is the probability that she is in dorm Y The probability is Tyne an integer or decimal rounded to three decimal places as needed CIES P Room X 0 7 0 4 P Y 0 6 Room 0 28 0 3 Apartment 0 12 02 Room 0 12 P Apartment Y 0 8 Apartment 0 48

TS shows an undirected weighted graph Rajah di bawah menunjukkan satu graf tak terarah dan berpemberat D 2 6 7 3 H E G 00 8 9 13 G 10 B 4p 25 The minimum total weight of the tree for the graph is 4p q The maximum total weigh tree for the graph is 4p q 2 Calculate the values of p and q Jumlah nilai pemberat minimum pokok bagi graf itu ialah 4p q Jumlah nilai pemberat mak pokok bagi graf itu ialah 4p q 2 Hitung nilai p dan nilai q 5 marks 5 ml chows 11 states except Federal Territory of Kuala 400 13

39 Sey eller Hint Okyanusu ndaki 115 ten fazla ada zerinde kurulu bir lkedir Sey eller bayra yapmak i in a a daki dikd rtgenin k sa ve uzun kenarlar n n zerine e it aral klarla iki er nokta i aretleniyor Bu noktalar dikd rtgeni 5 alana b lecek bi imde A k esiyle birle tiriliyor ve a a daki gibi boyan yor E A 15 C C 6 3 37 AB 3 37 birim AC 15 birim oldu una g re CE ka birimdir A 2 B 4 B D 6 5 E 9

Geometry For 25 28 tell whether the figure can be folded in half so that one side matches the other Yes No 28 27 26 25 S D

Write short answers Consider the inverse tangent function defined by y tan xory arctan x Complete parts a d a What is its domain Type your answer in interval notation Simplify your answer Type an exact answer using x as needed Use integers or fractions for any numbers in the expression b What is its range 00 00 Type your answer in interval notation Simplify your answer Type on exact answer using as needed Use integers or fractions for any numbers in the expression c Is this function increasing or decreasing O decreasing O increasing d is there any real number x for which arctan x is not defined If so what is it or what are they O Yes o bo Use a comma to separate answers as needed

Lesson 2 Practice Problems 1 Line SD is a line of symmetry for figure is congruent to HMZDS because sides AX and HM are corresponding a Why is Noah s congruence statement incorrect b Write a correct congruence statement for the pentagons X 2 Figure MBJKGH is the image of figure AFEKJB after being rotated 90 degrees counterclockwise about point K Draw a segment in figure AFEKJB to create a quadrilateral Draw the image of the segment when rotated 90 degrees counterclockwise about point K Write a congruence statement for the quadrilateral you created in figure AFEKJB and the image of the quadrilateral in figure M BJKGH A D B M E K

Given csc 2 Find all trigonometric function values for 8 with 8 in quadrant II Round all non termination decimals to three places sin a cos a tan a cut a sec a cac a

Macmillan Learning The mil is a type of angle measure used in the military This term is derived from milliradian and 1000 mils equal 1 radian What is the degree measure of 1 mil Use decimal notation Give your answer to three decimal places Include the degree symbol in the answer 1 mil A target is 1 yard wide and subtends an angle of 1 mil in a soldier s field of vision How far from the soldier is the target Suggestion Think of the target as an arc of a circle yards between the soldier and the target How many mils are there in a circle Use decimal notation Give your answer two decimal places

O Macmillan Learning Convert an angle of radian measure 6 to degree measure Give an exact answer Include the degree symbol in answer 67 radians