Geometry Questions

7 Parallelogram ABCD has coordinates of A 7 1 B 2 3 and C 0 3 What must be the coordinates of point D Explain how you found your answer y

Given that ABCD is a parallelogram with diagonals intersecting at M Segment PQ is drawn such that P lies on AB and Q lies on DC and it passes through M Prove that PM QM A P M B

1 In the diagram below ZDEF has been reflected in line t by reflecting points D E and F a As in class draw in segments DD EE and FF Mark the intersections of the segments with the line t b What is true about the points that you marked in part a Verify using an appropriate tool c What is true about the angles that the segments make with line t Mark these on your diagram D D E E tractor that the two angles are equal F

Given sin a 0 781 and sin 0 936 with both angles terminal rays in Quadrant l find the following Round your answer to 3 decimal places as needed tan a 8

6 A ladder leans against a building that is perpendicular to the ground such that it makes a 74 angle with the ground and its bottom lies 2 75 feet from the base of the building a Determine the height that the ladder reaches up the wall to the nearest tenth of a foot b How long is the ladder Round to the nearest foot 74 2 75 ft boooooo HH

If sin a 0 444 and sin 0 774 with both angles terminal rays in Quadrant l find the values of cos a 3 sin a 3 Your answers should be accurate to 4 decimal pla

5 An isosceles triangle has legs of length 12 and base angles that measure 28 What is the measure of its bas to the nearest tenth Be careful on this one

PROBLEM SOLVING 1 In each diagram shown below solve for the length of side CD using the information given Show the trig ratio you use and your algebra to find your final answer Round all answers to the nearest tenth b c a D 15 E E D 75 32 43 C d D 56 35 D 27 28 B B

4 In rectangle ABCD shown below CD 24 and m ABD 35 Determine the length of diagonal BD to the nearest tenth Show the work that leads to your answer A B 35

3 In AQRS ZR is a right angle m Q 57 and QS 35 Which of the following expressions correctly calculates the length of QR 1 2 35 sin 57 35 cos 57 3 35 cos 57 4 35 tan 57

Assume lies in quadrant 3 and the terminal side of 0 is perpendicular to the li y 19 x 6 Part 1 Determine sin e Part 2 Determine sec 8 sin 8 sec 8 X

cos u sin 2u cos 2u tan 2u 336 125 73 24 25 25 336 527 2 U X X

Given right triangle ABC shown below do the following a Measure the three side lengths to the nearest millimeter AB BC sin A b Based on your measurements in a state the value of each of the following in decimal form Round to the nearest thousandth cos A AC c Given the partial table of trigonometric values shown below which angle in the table is closest to the m A Verify using your protractor tan A Angle 37 38 39 40 41 A sine 0 6018 0 6157 0 6293 0 6428 0 6561 cosine 0 7986 0 7880 0 7772 0 7660 0 7547 B tangent 0 7536 0 7813 0 8098 0 8391 0 8693 0 0001

Students often struggle with the following statement All squares are rectangles but not all rectangles are squares Based on your knowledge of rectangles from this lesson and your knowledge of squares from middl school why is this statement true

4 Although it may seem obvious from the diagram of rectangle ABCD pictured ADMC is isosceles Explain why this must be true M

Given quadrilateral EFGH with vertices at E 4 8 F 8 4 G 5 5 and H 7 1 prove using coordinate geometry that EFGH is a rectangle Note that there are a few different methods that work y

10 Consider the definitions of the sine and cosine ratios Why can their values never be greater than 1 Use the triangle below as reference to help you answer C

A square has its length and width increased by two feet and four feet respectively so that it is now a rectangle The perimeter of the new rectangle is eight feet less than twice the original perimeter What was the original perimeter Show all work that leads to your answer Hint Let x square If you cannot do an algebraic approach at least attempt it by guess and check the side length of the original

4 Square ABCD has vertices at A 8 1 B 3 6 and D 3 10 What are the coordinates of point C TL

4 8 In the following diagram AEFG AMNP and cos E If MP 20 then find the length of MN to the 7 P nearest tenth Show the work that leads to your answer G S 20

4 Given that RSTU is a parallelogram and ST TU explain why RSTU must also be a rhombus R

C A rhombus has a perimeter of 80 inches Its longer diagonal is 32 inches Explain why the shorter diagonal must be 24 inches

In the following diagram of AQRS QR 8 and QS 17 Determine the value of sin Q Show how you arrived at your answer Q 17 8

3 In the diagram below of rhombus ABCD ADAC Which of the following is the measure of ZBDC A B 1 30 2 45 3 60 4 75 D E C

Construct a line AB such that it is parallel to RS shown below Use only a straightedge and compass Leave all construction marks Hint plot a third point not on the line and rotate about this point S

Given the rotation of AABC shown below determine the counter clockwise angle of rotation that occurred about point D Show all relevant lines and measurements you made to determine your answer A A C B C

E Using only a straightedge and a compass construct the image of AEFG after a rotation by 180 about point K shown Leave all construction marks Mark the image AE F G E K

5 Given the triangle below answer the following a Write trigonometric ratios based only on the information given for ZL and ZN ZL ratio ZN ratio b Solve both of these for the measures of ZL and ZN to the nearest tenth c What is the relationship between these two angles 10 5

I Given the right triangle shown below with m A 67 measure each side length to the nearest millimeter and then fill out the table For the ratios express your answers in decimal form to the nearest hundredth BC AB AC b cos cos 67 BC AC AB AC BC Making sure your calculator is in DEGREE MODE evaluate each of the following and verify that they either match or are close to the ratios you found in 1 a sin 67 c tan 67 AB A 67 C

3 Given right triangle DEF shown write down each of the following ratios Do not express as decimals a sin D b cos D c tan D d sin F f tan F e cos F 15 D 12 4 In any right triangle what must be true about the sum of the measures of the two acute angles Do you notice anything shout ho ging and cogine of the acute angles in 32 What do you notice

Joey wishes to contribute three items to a gift box He aims to pick from seven particular items Find the number of ways he can select the three items from these seven items Select one a 45 b 210 c 35 d 220

In trapezoid MNPQ shown below MN OP m M 108 m P 4x 15 and m Q 3x 21 Explain why m N must be 127 M N

7 Quadrilateral EFGH has vertices at E 1 8 F 6 1 G 4 4 and H 9 5 a Prove that EFGH is a parallelogram b Prove that EFGH is not a rhombus Many methods

5 In the diagram below MNPQ is a parallelogram whose diagonals are perpendicular Prove MNPQ is a rhombus M Q R N P

Given that CDEF is a trapezoid with CD FE list all pairs of supplementary angles C

Weights in grams of apples are normally distributed with a mean weight of 110 and a standard deviation of 5 5 grams An apple is selected at random Which has the greatest probability Select one a The apple weighs less than 75 grams b The apple weights between 95 and 100 grams c The apple weighs between 115 and 120 grams d The apple weights greater than 150 grams

The faces of a normal die are numbered from one to six Three such dice are used in an experiment The first one is thrown and the number shown on the top face is a five The second and third dice are then thrown Calculate as a fraction the probability that All three numbers shown are odd Select one a 2 3 b 1 2 Oc 1 6 d 1 4

Complete the equations which represent the relationship between polar and rectangular coordinates r sin tan x y Choose Choose Choose r 2 xr V 4

Rewrite the vector components using the unit vector form 7 13 3 U 13 3

6 Quadrilateral ABCD has vertices at A 0 6 B 4 1 C 4 0 and D 8 7 Prove that a ABCD is a rhombus using the distance formula b The diagonals of ABCD are perpendicular a

3 An isosceles triangle ABC has legs of length 24 and a vertex angle that measures 36 Determine the length of its base BC to the nearest tenth A 24 36 24

If 7 points are found on a circle how many triangles can be drawn using any 3 of these points as vertices Select one a 210 b 44 C 35 Od 38

Given the polar equation r 1 5sing Select one O a x 24y 30y 9 0 O b x 26y2 15y 9 0 Oc 24y x 15y 9 0 d None of these which is the rectangular equation

Graph the circle x2 y2 2x 8y 19 0 Ty Select one a 5 Ob 10 10 10 10 x

A rope is used to support a flag pole The rope is attached to the top of the 42 ft pole and attached to the ground 15 1 ft from the base of the pole How long is the rope Select one a 140 6 ft O b 44 6 ft OC 49 6 ft d 33 6 ft

If a 5 4 and ka 30 24 then k 9 8 7 6 5

A lighthouse is 46 m in height A boat s captain determines that the angle of elevation from his boat to the top of the lighthouse is 19 How far is the boat from the lighthouse Select one a 119 m b 145 m c 134 m d 186 m

A sailor is sailing around three islands She travels 2 6 km from Island A to Island B She then turns 55 and travels 2 9 km to island C How long is the last leg of the trip from Island C to Island A Select one a 1 3 km b 2 6 km OC 8 5 km d 2 9 km

Use your calculator to find the measure of the angle to the nearest degree tan 9 1

6 In trapezoid ABCD shown below AB DC BC LDC and sides have lengths as shown a Determine the measure of ZD to the nearest degree b Find the length of AD in simplest radical form D A 7 in 5 in B 3 in C