Solution of triangles Questions and Answers

Given a parallelogram ABCD with D lying inside the outer circle ABC Point E is the point of intersection of the outer circle ABC with line BD where E B Point F is the point of intersection of the outer circle AABC with line CD where F C It is known that AEC 55 and EAF 30 2 DAE is A 35 B 40 F E B C 45 D 50

Solve the right triangle with a 4 and c 10 State exact answers as discussed in the notes Remember use sin or cos to find the exact angles b Do not round until the final answer Exact only Do not round until the final answer Exact only B Do not round until the final answer Exact only

4 Write the conditional converse and contrapositive to the following inverse statement If a triangle has 1 right angle then the triangle is a right triangle 40 Conditional Converse Contrapositive 15 Draw a counterexample to the following statement All isosceles triangles are acute triangles 16 A toddler s climbing ramp is designed with the profile of a right triangle The measure of one acute angle is who times the measure of the other acute angle Find the measure of each acute angle Find m2BDC M 28 58 b 84 c 56 d 112 the 5 sides lengths from shortest to longest on the es below 8 7 951 135 B 8 19 A Geometry student was using the following picture to find mzQ Identify the first step if any in which an error occurred R 2x 15 a Yes acute triangle b Yes obtuse triangle c Yes right triangle d No cannot e a triangle X 2 20 Determine if the following measures can be the sides of a triangle 6 14 13 If so classify the triangle by its angles Side 21 Find the value of x Then classify the triangle by its sides angles Angle TO113 22 Find the measure of the smallest angle H 4x 2x 9 W Step 1 Step 2 Step 3 Step 4 10x 3x 8 23 Determine the value of x x x 200 Student Work x 2 2x 15 180 5x 30 180 C mZQ 2 30 15 75 x 150 x 30 3x 28 B 5x 52 2x 10

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTION 2 sin 9 cos 8 1 8

Triangle FGH is congruent to triangle QRS Which congruence statements must be true Select ALL that apply FG RS OZGHF LRQS HF SQ OLHGF SQR LCE

a sin 0 b cos 8 c tan 0 d cos 20 e tan 20 cot 20 g sec 20 h CSC 28 12

Use a double angle formula to rewrite the expression Use a graphing utility to graph both expressions to verify that both forms are the same 8 sin x 4

Determine whether the Law of Sines or the Law of Cosines can be used to find another measure of the triangle a 11 b 13 c 8 Law of Sines O Law of Cosines Then solve the triangle Round your answers to two decimal places A B C Mond 11 n O

25 mm 60 60 These figures are congruent What is men 50 mm 30

Using the prototype oblique triangle shown below and the information given find the measure of the remaining sides and angles in the triangle b a 7 C a If b 9 68542 y 124 and a 3 50173 then B

1 The shadecloth outside a workplace is 3 m long The building is 2 4 m high How far are the pegs from the base of the building 2 A city park is a square with sides 320 metres long The council decides to build a new walkway from one corner to the opposite corner How long will the walkway be B 3 2 m 14 2 m shadecloth 3 m 2 4 m building Find the length of the truss AB for the roof structure shown

Given the points below as the vertices of a quadrilateral ABCD determine which sides segments ar parallel to each other and which are perpendicular Hint Draw the shape and name each of the four sides then find the slope of each A 2 5 B 6 3 C 5 1 D 7 2

Given the following figure prove that the m 14 31 degrees 156 20 14 35

3 26 X SA

In the diagram below of triangle UVW X is the midpoint of UW and Y is the midpoint of VW If XY 6x 61 and UV 3x 17 what is the measure of UV U X W Y Show Examples V

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 3 sec 0 4 0 0 rad

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

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

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

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

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

If GH GJ m GIH 6b and mZGIJ b 60 what is mZGIJ G H J I

If JK 9w 67 and GI 2w 12 what is the value of w G H I

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

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

Given AXYZ ARST find the values of a and b N 5b 3 70 4a 4 R 62 T 48

20 tan 22 tan tane 90 Y 24 tan 0 X 0 270 90 undifined Ne saution 30 21 sin 0 23 cos 25 tan 780 780 360 510 2 Que V 120 costo x 1 360V 60 X 810 3 81c 8 10 TO 366 958 860 40 und

Solve the right triangle shown in the figure A B C a b C 56 10 B h

Find the value of x using the figure shown to the right The value of x is Simplify your answer Do not include the degree symbol in your answer 5 x 7

Solve each triangle ABC that exists A 36 1 a 3 1 c 10 7 Select the correct choice below and if necessary fill in the answer boxes within the choice O A There is only one possible solution for the triangle The measurements for the remaining angles B and C and side b are as follows B c b Round to the nearest tenth as needed OB There are two possible solutions for the triangle The measurements for the solution with the longer side b are as follows B 1 C b The measurements for the solution with the shorter side b are as follows B C Round to the nearest tenth as needed O C There are no possible solutions for the triangle

Question Solve for x Figures are not necessarily drawn to scale Answer C 3 F 2 D E 3 2 B X W

ASTU APQR Find PQ and SU T Q S 6 9 14 Write your answers as decimals or whole numbers PQ SU G 14 14

Find the value of the variable and YZ if Y is between X and Z XY 2c 1 YZ 6c and XZ 81 Just find the length of YZ

m 1 and m 3 m 2 and m 3 m 4 and m 5 m 5 and m 8 IIII I I

olem 5 Suppose that a species of fish in thousands is modeled by t 10 F t 0 5t2 1 120 where t is measured in years after 2000 a What is the initial population b Find F 10 and interpret its meaning c Find all horizontal asymptotes of F t if any d Sketch the graph of the function F t Label and appropriately scale the axes to include the ent graph e Evaluate the average rate of change of F t from 2004 to 2006 Interpret the meaning of y answer and include appropriate units f As time increases describe what happens to the fish population

Select the correct answer from each drop down menu In the given image line AB is parallel to line DC and DF BF D B By the alternate interior angles theorem ZFDCZFBA and by the vertical angles theorem ZAFBZC Complete the statement ADFC A by the SAS HL criterion Reset Next

Select the correct choices that complete the sentence below The value of tan 150 is exact value of tan 150 is because 150 is in quadrant The reference angle is

Consider a triangle ABC like the one below Suppose that a 53 b 6 and c 54 The figure is not drawn to scale Solve the triangle Carry your intermediate computations to at least four decimal places and round your answers to the nearest tenth If there is more than one solution use the button labeled or 0 B C 0 OF E X No solution

J B 74 3 a 859 m b 783 m

What is an angle An angle is I

Use the Law of Sines to solve if possible the triangle If two solutions exist find both Round your answers to two decimal places If a triangle is not possible enter IMPOSSIBLE in each corresponding answer blank A 139 a b 16 B C C

12 1 point D C 50 degrees 90 degrees Previous A If the measure of arc AB is 50 degrees what is the measure of angle ADB 25 degrees 100 degrees B

0 6 25 Points Determine two coterminal angles one positive and one negative for each angle Give your answers 19t 8 a DE b 8 0 rad 7T 15 rad

Find the measure of angle A 9 Fa 71x 1 11 0 x 51 20x x 61 80 A 28 28 10 A 12 x 34 otsib olgua does lo ozess x 64 x 52 75 67 x

20 A ray that divides the angle into two angles is the angle bisector 21 An obtuse triangle has exactly one angle whose measure is greater than 90 biosoqsil 22 Suppose a set of thin rods is glued together into a triangle as shown How would you place the triangular arrangement of rods onto the edge of a ruler so that they balance Explain why

3 Scientists measure the depths of lunar craters by measuring the length of the shadow cast by the edge of the crater using photos In a photograph the length of the shadow cast by the edge of the Moltke crater is about 606 metres given to the nearest metre The sun s angle of elevation is 65 Find the depth of the crater giving your answer rounded to the nearest metre

A triangle has a base length of 3 1 cm and corresponding height 4 2 cm correct to 1 decimal place Calculate the upper and lower bounds for the area of the triangle as accurately as possible 4 2 cm 3 1 cm

In the figure below g o Find the values of x and y g m 70 3y 26 x O X y