Geometry Questions

Find all solutions of the given equation Enter your answers as a comma separated list Let k be any integer Round terms to two decimal places where appropriate 4 cos 8 3 0 0 rad

he smallest positive number for which f x 2x 1 f cos f z 0

10 In AABC at the right CD is an altitude and CE is a median as shown a Which angle has a larger measure ZCDB or ZCEB b If the m B 72 m A 50 and mZAEC 112 then m ECD and m CEB c Is CD a median for ACEB WIND Explain str All Rights Reserved MathBits com A PAPE 180 90 18 MathBits com D B

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to two decimal places where appropriate sin 0 1 8 rad

Prove the identity cos 2x sin 2x cos 4x Use a Double Angle Formula and simplify cos 2x sin 2x cos 2 11

Prove the identity sin x cos x 1 sin 2x Expand the product and use a Pythagorean Identity and a Double Angle Formula to simplify sin x cos x sin x 2 sin x cos x 1 2 sin x

Find sin 2x cos 2x and tan 2x from the given information sin x 8 17 x in Quadrant III sin 2x cos 2x tan 2x

Simplify the expression by using a Double Angle Formula or a Half Angle Formula a 2 tan 4 1 tan 4 b 2 tan 40 1 tan 40

Prove the identity sin 10x 2 sin 5x cos 5x

A right triangle has legs with lengths of 28 centimeters and 20 centimeters What is the length of this triangle s hypotenuse in centimeters A 8 6 B 4 74 C 48 D 1 184 6 The perimeter of an isosceles triangle is 83 inches Each of the two congruent sides of the triangle has a length of 21 inches What is the length in inches of the third side 7 In a right triangle the measure of one of the acute angles is 51 What is the measure in degrees of the other acute angle A 6 B 39 19 D 31 8 Each side of a 30 sided polygon has one of three lengths The number of sides with length 8 centimeters cm is 5 times the number of sides n with length 3 cm There are 6 sides with length 4 cm Which equation must be true for the value of n A 5n 6 30 B 6n 6 30 C 8n 3n 4n 30 D 8 5n 3m 4 6 30 9 A right circular cone has a height of 22 centimeters cm and a base with a diameter of 6 cm The volume of this cone is n cm What is the value of n

Name Date Carefully read and respond to each question You will be taking the PSAT on Tuesday 10 24 2023 References 1 A 70 B 140 C 1 156 D 1 224 A C 2pr V fwh A w A 8 B 16 C 64 D 128 V nr h h What is the area in square centimeters of a rectangle with a length of 36 centimeters and a width of 34 centimeters 2 There are 170 blocks in a container Of these blocks 10 are green How many blocks in the container are green 3 The area of a square is 64 square inches What is the side length in inches of this square a The number of degrees of arc in a circle is 360 The number of radians of arc in a circle is 2 The sum of the measures in degrees of the angles of a triangle is 180 2x 60 2 30 4 A right rectangular prism has a length of 28 centimeters cm a width of 15 cm and a height What is the surface area in cm of the V h 2 45 S x 3 Special Right Triangles Perioa V v wh

October 20 A 3 2 7 9 S A un m N 6 5 4 3 2 1 1 2 45 B 5 6 3 4 5 6 7 8 9 10 11 12 13 14 15 P 0 2 4 4 C 13 2 D 5 2

The SAS or side angle side criterion states that two triangles are similar if two pairs of corresponding sides are in the same ratio and their included angles are congruent You ll use GeoGebra to demonstrate this Use the GeoGebra activity to complete each step below For help watch these short videos about using GeoGebra to draw points lines and angles and use the measurement tools Part A Create a random triangle AABC Record the lengths of sides BC and CA and the measure of the included angle Take a screenshot of your dilation save it and insert the image below the table BIUX X 10pt V AVA 2BCA side BC side CA Measure

You will use the GeoGebra geometry tool to investigate whether two pairs of corresponding angles in two triangles are enough to prove that the two triangles are similar Open the GeoGebra activity to complete each step below For help watch these short videos about using GeoGebra to draw points lines and angles and use the measurement tools Part A Create a random triangle AABC Measure and record its angles BIX X 14pt v AVA Angle Measure ZCAB ZABC ZACB Space used includes formatting 803 15000 Part B Now you will attempt to copy your original triangle using two of its angles Draw a new line segment DE of any length and parallel to AB from AABC You can draw DE anywhere on the coordinate plane but not on top of AABC From point D create an angle with the same measure as ZCAB From point E create an angle with the same measure as ZABC Draw rays from points D and E to E and D respectively to form the angles Label the point of intersection of the two rays F Draw ADEF by creating a polygon through points D E and F K Record the measures of the angles of ADEF Take a screenshot of your results save it and insert the image below the table BIY X X 14pt Angle Measure EDE AVA S A

O Find the value of x 16x 4 9x 6 58

If we want the median of these five numbers to be 45 what number belongs in the blank If nothing will work just type NOT POSSIBLE 25 90 10 60

Find the value of x and all missing angles Then classify the triangle by the angles 2x 3x 55 5 P 3x 3x 20 9x 5 S R 6 55 60

1 Points DETAILS LARPRECALCRMRP7 5 2 072 Powers of trigonometric functions are rewritten to be useful in calculus Verify the identity Simplify your answers completely sec x tan x tan x tan x sec x tan x tan x sec x Need Help Read It Submit Answer 10 1 Points S DETAILS tan6 x sec x tan x h ft Gnomon The lengths of a shadow cast by a vertical gnomon a device used to tell time of height h when the angle of the sun above the horizon is 8 see h sin 90 0 sin 0 5 Watch It a Verify that the expression for s is equivalent to h cot 0 Use the Cofunction Identity Simplify your answer completely h sin 90 0 sin 0 h cot 0 0 1 tan x sec x LARPRECALCRMRP7 5 2 079 sin 0 b Use a graphing utility to complete the table Let h 4 feet Round your answers to two decimal places 15 30 45 60 75 90

Rewrite the expression as a single logarithm and simplify the result Hint Begin by using the properties of logarithms 7 In cott 7 In 1 tan t

Fill in the blank to correctly complete the If a function f has an inverse then the graph off may be obtained by reflecting the graph of f across the line with the equation Type an equation

A telephone line hangs between two poles 14 m apart in the shape of the catenary y 22 cosh 7 5 0 0 0 7 x 2 17 where x and y are measured in meters 22 a Find the slope of this curve where it meets the right hand pole Round your answer to four decimal places b Find the angle 8 in degrees between the line and the pole Round your answer to two decimal places 8

angle me Jane and Opal s straw designs knowing that the lines created by the straws in their designs were parallel 70 Jane s straw design 135 140 xo to 70 Opal s straw design Find all of the labeled angle measurements knowing that the lines created by the straws are parallel Explain the reasoning for your results

Each one was making a line design using either 3 or 4 straws They had just come from math class where they had been studying special angles Paul pulled his pencil out of his book bag and labeled some of the angles and lines He then challenged himself and the others to find all the labeled angle measurements in Paul and Justin s straw designs and to determine whether the lines that appear to be parallel really are parallel Paul s straw design 2C B 40 20 2x 10 5x 20 Justin s straw design began playing with drink straws 3x 30 Find all of the labeled angle measurements Determine whether the lines that appear to be parallel really are parallel Explain the reasoning for your results

Find the number of integer values of BC x 8 A D

Use a graphing utility to determine which of the six trigonometric functions is equal to the expression Verify your answer algebraically 1 1 sin 0 cos 8 cos 0 1 sin 0

Factor the expression and use the fundamental identities to simplify Use a graphing utility to check your result graphically sec r tan x sec r STEP 1 Multiply constants sec x tan x sec x 16 STEP 2 Factor sec x tan x 6 sec x 16 sec x tan x sec x 16 tan x 1 STEP 3 Apply the appropriate fundamental trigonometric identity and simplify 7 sec x tan x sec x 16

Use the values to evaluate if possible all six trigonometric functions If an answer is undefined enter UNDEFINED tan 0 is undefined sin 0 0 cos 0 sin 0 csc 0 sec 8 cot 0

Complete the equation for the line perpendicular to g x 6x 5 and passing through the point 9 2 help equations Y

a Find the domain of f The domain of f is 0 00 Type your answer in interval notation b Choose the correct graph of f A Ay B O C Q O D

Describe a pattern in the sequence of numbers Predict the next number 5 2 6 18 54 6 0 1 4 9 7 256 64 16 4 8 3 0 3 0 3 0 9 7 0 7 5 8 0 8 5 10 13 7 1 5

Sketch the next figure in the pattern 3 ADO 4

1 Explain what a conjecture is 2 How can you prove that a conjecture is false Sec 10 F

A parcel delivery service has contracted you to design a closed box with a square base that has a volume of 11 000 cubic inches Complete parts a d a Express the surface area of the box as a function of x X

2 2x 3 For the function F x X domain and range and c use the final graph to list any vertical horizontal or oblique asymptotes a graph the rational function using transformations b use the final graph to find 17 2 a Which of the following transformations is required to graph the given function 1 OA Shift the graph of y 3 units to the left and shift it 2 units up 2 x O B Vertically stretch the graph of y by a factor of 3 reflect it about the x axis and shift it 2 units down X OC Vertically stretch the graph of y by a factor of 3 reflect it about the y axis and shift it 2 units up X Vertically stretch the graph of y hy a factor of 3 reflect it about the x avis and shift it 2 units un

Use the graph shown to find the following a The domain and range of the function b The intercepts if any c Horizontal asymptotes if any d Vertical asymptotes if any e Oblique asymptotes if any MIDD 9 6 3 Q Find the vertical asymptotes if there are any Select the correct choice below and fill in any answer boxes within your choice O A x Type an integer or a simplified fraction Use a comma to separate answers as needed OB There are no vertical asymptotes Find the oblique asymptotes if there are any Select the correct choice below and fill in any answer boxes withi

For the function H x x 2 a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertica horizontal or oblique asymptotes a Choose the correct graph below O A O B G C O B The domain of the given function is xlx is a real number x Type an integer or a simplified fraction O C M O D b What is the domain of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice O A The domain of the given function is xlx is a real number x Type an integer or a simplified fraction Q

4 For the function H x a graph the rational function using transformations b use the final graph to find the X 2 domain and range and c use the final graph to list any vertical horizontal or oblique asymptotes a Choose the correct graph below O A B Q O C Q D

Use the graph shown to find the following a The domain and range of the function b The intercepts if any c Horizontal asymptotes if any d Vertical asymptotes if any e Oblique asymptotes if any C 9 6 Find the y intercepts if there are any Select the correct choice below and fill in any answer boxes within your choice O A y Type an integer or a simplified fraction Use a comma to separate answers as needed

Charles RYE Bushels Dina On the following graph use the blue line circle symbol to plot Charles s production possibilities curve PPF and use the purple line diamond symbol to plot Dina s PPF 200 180 160 140 120 100 80 60 40 20 0 4 Corn Bushels per acre 40 28 0 100 Rye Bushels per acre 8 7 200 300 400 500 600 CORN Bushels 700 800 900 1000 Charles s PPF Dina s PPF

Graph the function by setting the x intercept and the asymptotes Provide your answer below 10 2 0 5 10 3 0 f x P 4 10

N 7 Figure 1 Map of 300 mb winds at 12Z 10 MAR 2011 1 Look at the accompanying Figure 1 With a pencil shade a small oval around each station where wind speeds are 60 knots or greater such as San Francisco Then shade the area between adjacent shaded stations where the wind speeds are also presumed to be 60 knots of more Next draw lines along the northern and the southern boundaries to highlight the shaded area limits of the 60 knot or greater region Finally draw a heavy smooth curved arrow through the core of the shaded highest wind speeds Add an arrow head to show wind direction 2 The large arrow you drew on your map approximates the location of the existing polar front jet stream across the United States Now imagine that you are in a gondola attached to a helium filled balloon that is located over central Washington State Assuming your balloon stays at the 300 millibar level and travels with the air flow draw your path as you travel across the country Through what state or states are you likely to leave the United States

15 Solve a system of equations to find the values of x and y Show all work y 30 4x 3x 2y 9 0 waled sugit ani or

16 Given c d and 3 24 Prove 42 24 Statement Reason

D C b G A F Refer to the figure index the following vectors and planes Vector AB AE and BD Plane AFEC FBCD and BFGE E B

Complete the two column proof 17 Given l 2 and 21 23 Prove j k Statement 12 2 1 Es bns b305 3 11m91512 Reason

Based on the diagram determine if the statement is True or False 1 2 3 4 5 6 7 28 218 are alternate interior angles 214 223 are same side exterior angles 414 220 42 26 24 210 23 216 are supplementary 222 224 Call 12 4 3 S6 8 13 14 1615 9 10 11 12 17 18 20 19 21 22 24 23

Based on the diagram determine whether each 48 46 1 132 46 132 8 9 10 1 11 12 cll f cll d clle dlle all b

2 If mLABD 39 and m2DBC 22 what is mLABC Draw two angle pairs that will justify your answer

R B 90 c 106 02 D Transversal CD cuts parallel lines PQ and RS at points X and Y as shown in the diagram If m2CXP 106 02 what is m SYD OA 73 98 OD 180 S

Question 10 Write a logarithmic equation corresponding to the graph shown The dashed line indicates the asymptote of the function Use y log x as the parent function 7 6 V 4 3 2 1 7 6 5 4 3 2 1 1 2 3 4 5 6 7 X 12 3 4 5 6 7