2D Geometry Questions and Answers

Finding Arc Length If e 15 what is the length of AB 0 S S S S 360 angle whose 24 7 12 147 T 2 r 2 Radian Measure One radian is the measure of a central is the t As B 70 A 0 1 rad S r

If the central angle has a measure of 1 radian it will subtend an arc that has the same length as its radius regardless of the size of the circle 0 Or 0 A 2 B s 0r When finding arc length use s 8r O D CE R 0 A R Finding Arc Length with a Central Angle Measured in Radians In circle O 0 2 6 radians What is the length of minor arc DE S S B S 31 2 cm

Circumference and Area of a Circle The circumference of a circle is the distance arc length s is the distance C 2 r 2 TT 31 4 units 3 14 is an approximation for 7 B MAB C 360 107 60 360 31 4 52 units the entire circle while two points on the circle 60 5 The arc length is as long as the entire circumference because the arc 6 represents of the circle N S B

The area of a circle is determined by the radius of the circle with the formula The area of a creating the sector Area of sector Area of sector dimens 0 360 is determined by the size of the central angle 0 when the central angle is measured in when the central angle is measured in

Sectors of Circles A sector of a circle is the region of a circle bounded by two intercepted Calculating Area of a Sector The area of a How large the central angle is in relation to the measure of the entire circle determines how big the sector is in relation to the entire circle The A The area of the B T C x Area of a sector r 360 of a circle is found by multiplying these two factors of the central angle over 360 degrees circle B and their 135 16

When a arc or different congruent arcs the central angle an angle will be supplementary angles mAOB m2ACB 180 1 angle and circumscribed angle intercept the Solving for Unknown Measures What is the measure of angle D The arc intercepted by an inscribed angle is twice the measure of that angle TR B 43 x 2 86 The measure of the central angle is the same as the arc it intercepts m BOC 86 mz The circumscribed angle is supplementary to the central angle 180 m m AOB mZWYX ST W 43 0 X B CO www 180 B 86 D O

38 1 4 km A 6 km C 8 6 km 2 6 km 2 km 2 B 8 6 km D 6 8 km

and tangents to a Compare lengths of of a circle Compare of a circle Words to Know Fill in this table as you work through the lesson You may also use the glossary to help you an angle whose vertex is on a circle and whose sides are chords a segment with both endpoints on a circle an angle whose vertex is at the center of a circle and whose sides are radii of that circle a segment of a tangent that has an endpoint at the point of tangency an angle whose vertex is outside of a circle and whose sides are tangents to that circle two angles whose measures have a sum of 180

When an inscribed angle intercepts a inscribed angle create a triangle The central angle is a straight angle so it has a measure of 180 The inscribed angle intercepts the same arc as the central angle so the measure of the inscribed angle is of 180 or 90 2 When the sides of the inscribed angle are congruent they create a 45 45 90 triangle the central angle and B Quadrilaterals Created by Central and Inscribed Angles GENERAL CASE What polygon is created when the inscribed angle does not intercept a diameter of the circle B Draw tick marks to show that the radil are congruent

40 10 m 8 m A 5 m C 12 m 6 m B 24 m D 14 8 m

What polygon is created when the Inscribed angle does not intercept a diameter of the circle N If the chords are congruent the quadrilateral formed is a kite The diagonals are The angles formed by the two non congruent sides are The diagonal that runs from the vertices of the two non congruent sides will be bisected 1

Relax How are the measures of inscribed angles and central angles related when they intercept the same arc or congruent arcs The Intercepted arc has the same measure as the central angle N The measure of the inscribed the intercepted arc will be half the measure of When a mzACB 1 of the 50 An inscribed angle that intercepts the same arc as a central angle will be of the measure of the central angle Determining Measures of Inscribed and Central Angles mAB mZAOB O congruent arcs the measure of the central angle will be angle 2 m AJB 0 angle and inscribed angle intercept the same arc or mZQOM 2 mZPRS O B P the measure

Relating Central and Circumscribed Angles UNDERSTANDING THE RELATIONSHIP How are the measures of circumscribed angles and central angles related when they intercept the same arc or different congruent arcs The intercepted arc has the same measure as the central angle THE The entire arc of a circle is 360 1 2 0 250 The measure of the circumscribed angle is half the of the intercepted arcs mADB mAB 110 70 D 0 110 B C

If the perimeter of kite NMOL is 42 units what is the length of side NL What is the length of diagonal NO 42 9 This is the combined length of NM and NL so divide by 2 NL 12 The point of tangency between a radius and a tangent segment is a right angle We can use the Pythagorean theorem to find NO NO 12 9 144 N 12 Angit M 9 O

What polygon is created by central and circumscribed angles that intercept the same arc The quadrilateral is a The radii and the tangents create right angles 6 The diagonals are perpendicular One diagonal is bisected H N M

Find the area of each 39 2 8 mi 3 3 mi A 12 24 mi C 18 48 mi 6 4 mi 3 3 mi B 9 24 mi D 4 6 mi

D PROVING THEY ARE Given Angle ACB is a circumscribed angle that intercepts the same arc as central angle AOB Prove m2ACB m2AOB 180 0 SUPPLEMENTARY B L m20AC m 0BC Tangents are perpendicular to radii at the point of tangency O m20AB mZ0BC m2AOB m2ACB The sum of the interior measures of a quadrilateral is 360 Subtract the measures of the two right angles m2ACB m2AOB 360 90 90 m2ACB m2AOB O

Find the peremiter of each 37 10 ft 6 1 ft 7 ft A 26 ft C 23 1 ft 9 ft B 25 1 ft D 26 ft

2 10 3 ft 9 2 ft 9 2 ft A 94 76 ft C 88 06 ft 10 3 ft B 189 52 ft D 47 4 ft

Central Inscribed and Circumscribed Angles Central angles circumscribed angles and inscribed angles are created tangent segments W W and outside of a circle by A chord is a segment that has two on the circle A tangent segment has an endpoint at the point of tangency that are Central angles are created by two radii They have a vertex at the of the circle The endpoints are on the circle and the sides are congruent radii chords and

Identify the and radius of a circle center Examine equations of circle hypotenuse radius given in standard form or general form Write the of a circle Words to Know Write the letter of the definition next to the matching word as you work through the lesson You may use the glossary to help you Determine if a given lles on a circle A the set of all points in a plane that are a given distance away from a given point called the center B a segment that extends from the center of a circle to any point on the circle C the fixed point that is equidistant from all points on a circle D the side of a right triangle that is opposite the right angle and is always the longest side of the triangle

The form of an equation of a circle is x h y k r The point h k is the center and is the The general form of an equation of a circle is x y Cx Dy E 0 1 Complete the form to convert from general form to standard

Identifying the Center and Radius Given an Equation in General Form Identify the center and radius of a circle whose equation is x y 10x 14y 58 0 Complete the square for the x and y terms Group together the x terms x 10x y 14y Determine what value needs to be added to both sides of the equation to complete the square y 14y x 10x 58 25 49 16 Factor the polynomials Identify the center and radius x y ap 1 71 214 X 2 2 19 2 25 14 24 y 49

Determining Whether a Point Lies on a Circle The point 4 0 lies on a circle that is centered at the origin Does the point 2 12 also lie on the circle d x x y y All radii of a circle have the same length r 4 R Is 2 12 4 units from the center x y 0 x2 y 2 4 2 0 12 0 16 Yes the point does lie on the circle 5 5 4 3 2 1 4 2 0 0 1 2 7 5 y 1 2 3 31 5

10 29anoq o bobna squ 12 2 points On Saturday July 1st Yellowstone National Park had 25 000 visitors in the park at 1 00 PM The total area of the park is 8 983 13 km and the total area of water is 2 714 13 km Determine the density of people on land in Yellowstone National Park on Saturday July 1st at 1 00 PM to the nearest visitor

The standard form of an equation of a circle centered at the x y r where r is the radius The hypotenuse of the triangle is a length is r Find the lengths of the legs The 1 PS y Substitute into the Pythagorean theorem y r where r is the Identifying the Radius and Center from an Equation h k 1 0 x 3 y 4 25 4 of the circle so its Q 0 0 and h k is the center of the circle form of an equation of a circle is x h y k r 2 x y 7 19 h is the value subtracted from x is the value Is T 19 h k 0 P x y S x 0 from

bubble for the letter corresponding to the answer you think choose an answer by mistake erase your mistaken shading thoroughly i to bio of 1 The figure below is vertically cut What is the cross section created A Circle B Triangle C Rectangle D Pentagon bebola sat od srld mi sevil sigos 20 30 40 50

an Equation with a Givel Center and Radius Determine the equation of a circle with center 7 6 and a radius of 4 units h k h k Writing an quation Given a Graph What is the equation of the circle shown in the graph x h y k r x 7 6 x 7 x h y k x 3 y 2 x 2 2 16 5 3 2 1 NWU 4 2 1 1 A 1 2 S

Pythagorean theorem In a right triangle the square of the length of the hypotenuse is equal to the sum of the of the lengths of the A C a B The hypotenuse is the longest side in a right triangle c a b

3 Jerome paints the box below in art class One can of paint covers 118 in How many cans of paint does Jerome need to buy A 1 B 2 C 3 D 4 simon 8 in 5 in 6 in

LI SB Use relationships with length area and volume when an object is dild 3 SB 3 A superhero action figure has a volume 1500 cm The figure is dilated to create a store display The volume of the dilated figure is 45 000 cm What was the approximate scale factor of the dilation 5B 4 A pet supply company makes a small bag that holds 2 pounds of dog food They want to make a bag for large dogs that holds 40 pounds of dog food By approximately what factor will the surface area of the bag increase Round your answer to the nearest tenth

33 2m 2 x 80 A 10 C 7 40 2 B 11 D 8

36 Find m ROS K S 24x 1 8 15x 47 Q A 145 C 100 R B 98 D 117

35 F120 20x 2 H A 12 C 9 11x 2 B 4 D 13 G

31 12 x 18 A isosceles 8 B equilateral 11 C isosceles 6 D equilateral 10

32 6 x 18 A equilateral 12 B isosceles 11 C isosceles 10 D equilateral 9

26 H A HL B ASA C LL D Not enough information

30 2r 5 HH H 15 7r A 4 SSS B 1 2 ASA SAS D 4 AAS

28 AAS L C K A MK LC B LK LC C KL D LM M CM or MK LC ML

L M K W A LK WX B KM XV C ZL W or K LX D ZL ZW V

25 A LA B HL C ASA D Not enough information

24 A AAS B Not enough information C HL D ASA

23 A Not enough information B SAS C AAS D ASA

18 E F A AWUV AUFE B AWVU AFUE C AUVWAUEF D AUWVAFEU W

19 K IJ A SU C UT AUTS B TS D ZU T S

16 G A rotation 90 clockwise about the origi B translation 2 units right and 5 units u C rotation 180 about the origin reflection across y x

12 rotation 90 clockwise about the origin D 2 4 C 2 5 B 3 5 A 2 1 A D 2 4 C 2 5 B 3 5 A 2 1 B D14 2 C 5 2 B 5 3 A 1 2 C D 4 2 C 5 2 B 5 3 4 1 2 D D 0 0 C 0 1 B 5 1 A 4 3

14 W V W X X X A rotation 90 clockwise about the origin B translation 2 units right and 6 units up C reflection across x 1 D translation 7 units right

D 10 rotation 180 about the origin A B F F Ay G y x

f x is shown in blue Draw the graph of y 2f x shown in red by determining how a stretch or compression will transform the location of the indicated blue points Drag the movable red points to the desired coordinates Provide your answer below 104 9 8 8 7 2 6 3 4 9 5