# Geometry

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Figure ABCD is a parallelogram. What is the value of p?
6
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Figure LMNO is a parallelogram. What is the value of x?
20
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Figure CDEF is a parallelogram. Which measures are correct? Check all that apply.
n = 10 CF = 59 FE = 42
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Figure ABCD is a parallelogram. What are the measures of angles B and D?
A. ∠B = 55°; ∠D = 55°
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Figure PQRS is a parallelogram. The expressions represent the measures of the angles in degrees. What is the value of x?
20
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Figure JKLM is a parallelogram. The measures of line segments MT and TK are shown. What is the value of y?
7
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The parallelogram shown represents a map of the boundaries of a natural preserve. Walking trails run from points A to C and from points B to D. The measurements shown represent miles. What is the sum of the lengths of the two trails?
16 miles
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A plot of land is in the shape of a parallelogram with the dimensions shown. How many feet of fencing are needed to enclose the lot?
68 feet
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Figure ABCD is a parallelogram. What are the measures of angles B and C?
B. ∠B = 65°; ∠C = 115°
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Figure ABCD is a parallelogram. What is the perimeter of ABCD?
44 unit
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What is the perimeter of parallelogram LMNO?
80
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The perimeter of parallelogram ABCD is 46 inches. What is DA?
4
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What is the measure of angle O in parallelogram LMNO?
105
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In parallelogram ABCD, what is DC?
13
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In parallelogram PQSR, what is PQ?
9
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Given: AB ≅ CD and AD ≅ BC Prove: ABCD is a parallelogram. What is the missing reason in step 3?
SSS congruency theorem
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In parallelogram LMNO, what are the values of x and y?
x = 55, y = 14
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What is the measure of angle L in parallelogram LMNO? m∠L =
40
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Which best explains if quadrilateral WXYZ can be a parallelogram?
WXYZ is not necessarily a parallelogram because it is unknown if WC = CY.
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Given: AD ≅ BC and AD ∥ BC Prove: ABCD is a parallelogram. What is the missing reason in step 6?
CPCTC
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Consider the diagram and proof below. Given: WXYZ is a parallelogram, ZX ≅ WY Prove: WXYZ is a rectangle What is the missing reason in Step 7?
consecutive ∠s in a ▱ are supplementary
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A rectangle has a width of 9 units and a length of 40 units. What is the length of a diagonal?
41 units
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Rectangle PQRS is shown with its diagonals, PR and QS. What type of triangle is △STR?
isosceles triangle
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Figure ABCD is a parallelogram. If ABCD is also a rhombus, what must be the value of x?
13
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What is the approximate side length of the square?
4.2 units
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Rhombus LMNO is shown with its diagonals. The length of LN is 28 centimeters. What is the length of LP?
14
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ABCD is a square. What is the length of line segment DC?
13
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A quilt piece is designed with four congruent triangles to form a rhombus so that one of the diagonals is equal to the side length of the rhombus. Which measures are true for the quilt piece? Check all that apply.
a = 60° The perimeter of the rhombus is 16 inches. The length of the longer diagonal is approximately 7 inches.
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The figure shown is a rhombus. Which equation is true regarding the angles formed by the diagonals and sides of the rhombus?
x + y = z
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The perimeter of the rectangle is 146 units. What is the length of the longer side?
45 units
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Rectangle PQRS is shown with its diagonals, PR and QS. What type of triangle is △STR?
isosceles triangle
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TREC is a rectangle. What is the length of ET?
34 units
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Which statements are true of all squares? Check all that apply.
The diagonals are perpendicular. The diagonals are congruent to each other. The diagonals bisect the vertex angles. The diagonals bisect each other.
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The figure is a square. Its diagonals meet to form four right angles. What is the approximate value of x?
5.7
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The figure is a parallelogram. One diagonal measures 28 units.
No, it is not a rectangle because the sides of the parallelogram do not meet at right angles.
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Consider kite ABCD. What are the values of x and y?
x = 5, y = 22
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Right trapezoid WXYZ is shown. What is the measure of angle WXY?
107°
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If KM is drawn on this quadrilateral, what will be its length?
18 units
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A kite has a perimeter of 70 centimeters. One of the shorter sides measures 16 centimeters. What are the lengths of the other three sides?
16 centimeters, 19 centimeters, and 19 centimeters
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B
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Given: JKLM is an isosceles trapezoid, KL ∥ JM Prove: KM ≅ JL What is the missing reason in step 4?
base angles theorem
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A kite has a perimeter of 108 feet. One of the longer sides measures 30 feet. What are the lengths of the other three sides?
24 feet, 24 feet, 30 feet
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If quadrilateral PQRS is a kite, which statements must be true? Check all that apply.
QP ≅ QR PM ≅ MR ∠QPS ≅ ∠QRS
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The diagram shows quadrilateral MNPQ. What is the length of line segment MQ?
11 units
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A town planner wants to build two new streets, Elm Street and Garden Road, to connect parallel streets Maple Drive and Pine Avenue. In trapezoid EFGH, EF ≅ HG. What is the measure of the angle between Elm Street and Pine Avenue?
72°
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Which statement proves that quadrilateral JKLM is a kite?
LM = JM = 3 and JK = LK = .
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In the diagram, WZ = square root 26 The perimeter of parallelogram WXYZ is +
8 units.
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The vertices of a quadrilateral in the coordinate plane are known. How can the perimeter of the figure be found?
Use the distance formula to find the length of each side, and then add the lengths.
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In the diagram, DG ∥ EF. What additional information would prove that DEFG is an isosceles trapezoid?
DE ≅ GF
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In the diagram, KL = 2 square root 2 , LM = square root 5, and MN = square root 2 . What is the perimeter of isosceles trapezoid KLMN?
C. 3 square root 2 + 2 square root 5 units
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In the diagram, AB = 10 and AC = 2 square root 10. What is the perimeter of △ABC?
D. 20 + 2 square root 10 units
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HIJK is a parallelogram because the midpoint of both diagonals is which means the diagonals bisect each other.
(1,0)
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Which statement proves that parallelogram KLMN is a rhombus?
The slope of KM is 1 and the slope of NL is -1.
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Which statement proves that the diagonals of square PQRS are perpendicular bisectors of each other?
The midpoint of both diagonals is 4 1/2 , 5 1/2 the slope
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What is the perimeter of △LMN?
8 + square root 10 units
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Given: ABCD is a kite. Prove: BD bisects AC. What is the missing reason in step 5?
corresponding parts of congruent triangles are congruent
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Kite ABCD represents a softball field that is being built. If AC = 48 meters, what is the perimeter of the field?
140 meters
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Maggie puts together two isosceles triangles so that they share a base, creating a kite. Each leg of the upper triangle measures 41 inches and each leg of the lower one measures 50 inches. If the length of the base of both triangles measures 80 inches, what is the length of the kite's shorter diagonal?
39 inches
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What is the perimeter of square ABCD?
B. 4 square root 37
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Figure ABCD is a square. Prove BD ≅ AC. What is the missing reason in the proof?
all sides of a square are congruent
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The diagram shows the shape of a plot of land that Maria will use for her garden. She needs to know the length of each side so she can buy fencing. What is the length of line segment FJ?
42 feet
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A 15-meter by 23-meter garden is divided into two sections. Two sidewalks run along the diagonal of the square section and along the diagonal of the smaller rectangular section. What is the approximate sum of the lengths of the two sidewalks, shown as dotted lines?
38.2 m
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Rhombus LMNO is shown with its diagonals. Angle MLO measures 112°. What is the measure of angle MLP?
56
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What is the perimeter of parallelogram WXYZ?
B. 2 square root 5 + 2 square root 17 units
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Square ABCD and isosceles triangle BUC are drawn to create trapezoid AUCD. What is the measure of angle DCU?
135o
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The perimeter of a rhombus is 28 centimeters. What is the length of each side of the rhombus?
7
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The diagonals of a parallelogram are congruent. Which could be the parallelogram?
rectangle
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In parallelogram WXYZ, WY = 12 in., XZ = 16 in., WX = 10 in., and XY = 9 in. What is the length of line segment MX?
8
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Given: RT ≅ TV and ST ≅ TU Prove: RSVU is a parallelogram.
△RTS ≅ △VTU and △RTU ≅ △VTS
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Jacob is cutting a tile in the shape of a parallelogram. Two opposite angles have measures of (6n − 70)° and (2n + 10)°. What are the two different angle measures of the parallelogram-shaped tile?
50° and 130°
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In circle D, which is tangent to the circle?
AB
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Which equation can be used to find the measure of ?
m:EHG + 80 + 35 = 360
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What does CDE most specifically describe?
a major arc
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Segments AC and BD are diameters of circle O. What is the measure ADE of ?
253
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In circle D, which is a secant?
GH
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Which is true regarding chords and diameters of circles?
Both chords and diameters have two endpoints on a circle. Diameters must intersect the center of a circle.
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In circle O, SU is a diameter. What is m:ST?
130
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What is the measure of CED?
212
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In circle O, RT and SU are diameters. If m:RV = m:VU, what is mVU?
64
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The measure of BAC is 240. What is the ratio of the measure of the major arc to the measure of the minor arc?
2:1
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Line segment XY is tangent to circle Z at point U. If the measure of UV is 84°, what is the measure of ?
42
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Angle KJL measures (7x - 8)o. Angle KML measures (3x + 8)o. What is the measure of arc KL?
40
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Given: Circle M with inscribed angel KJL and congruent radii JM and ML Prove: m angel MJL = 1/2 { m: KL} What is the missing reason in step 8?
substitution property
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In circle V, angle WXZ measures 30°. Line segments WV, XV, ZV, and YV are radii of circle V. What is the measure of WUX in circle V?
120
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Line segment GJ is a diameter of circle L. Angle K measures (4x + 6)°. What is the value of x?
21
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Line EF is tangent to circle G at point A. If the measure of angel CAE is 95°, what is the measure of CBA ?
190
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Line segment BD is a diameter of circle E. What is the measure of BC ?
78
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In circle D, angle ADC measures (7x + 2)°. Arc AC measures (8x - 8)°. What is the measure of angel ABC ?
36
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What is the measure of arc EF in circle H?
41
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Given: Circle O with diameter LN and inscribed angle LMN Prove: LMN is a right angle. What is the missing reason in step 5?
inscribed angle theorem
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RT and GJ are chords that intersect at point H. If RH = 10 units, HT = 16 units, and GH = 8 units, what is the length of line segment HJ?
20
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What is the length of line segment XZ?
16
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AX and EX are secant segments that intersect at point X. What is the length of DE?
3 units
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Circle C is inscribed in triangle QSU. What is the perimeter of triangle QSU?
40
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Line segment BA is tangent to the circle. What is the length of line segment BA? Round to the nearest unit.
98
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The circle is inscribed in triangle AEC. Which are congruent line segments? Check all that apply.
EF and ED CB and CD
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What is the length of line segment SV?
16 units
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Point O is the center of the circle. What is the perimeter of quadrilateral DOBC
28 units
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Line segment QP is tangent to the circle. What is the length of line segment QP? Round to the nearest unit.
20 units
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SU and VT are chords that intersect at point R. What is the length of line segment VT?
13 units
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What is the area of the shaded sector of the circle?
B. 27 r units2
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Line segment WX is the radius of circle X, and line segment ZY is the radius of circle Y. Points W, X, C, Y, and Z are all on line segment WZ. What is the area of circle C, which passes though points W and Z?
324 r units
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The measure of central angle XYZ is 1.25 radians. What is the area of the shaded sector?
40
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The diagram shows one way to develop the formula for the area of a circle. Pieces of a circle with radius r are rearranged to create a shape that resembles a parallelogram. Since the circumference of the circle can be represented by 2πr, and the area of a parallelogram is determined using A = bh, which represents the approximate area of the parallelogram-like figure?
C.
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Point G is the center of the small circle. Point X is the center of the large circle. Points G, H, and X are on line segment GX. What would be the area of a new circle that has line segment GX as its diameter?`
49
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Fiona draws a circle with a diameter of 14 meters. What is the area of Fiona's circle?
49
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The measure of central angle RST is radians. What is the area of the shaded sector?
8
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Which statements are true about circle Q? Check all that apply.
The ratio of the measure of central angle PQR to the measure of the entire circle is 1/8. The area of the shaded sector depends on the length of the radius. The area of the shaded sector depends on the area of the circle.
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The smallest of the three circles with center D has a radius of 8 inches and CB = BA = 4 inches. What is the sum of the areas of all three circles?
464
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In circle V, r = 14ft. What is the area of circle V?
196
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Points E, F, and D are on circle C, and angle G measures 60°. The measure of arc EF equals the measure of arc FD.
∠EFD ≅ ∠EGD ED = FD mFD = 120°
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Major arc JL measures 300°. Which describes triangle JLM?
equilateral
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Line segment ON is perpendicular to line segment ML. What is the length of chord ML?
24 units
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Angle G is a circumscribed angle of circle E. Major arc FD measures 280°. What is the measure of angle GFD?
40°
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Angle BCD is a circumscribed angle of circle A. What is the measure of minor arc BD?
100°
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Angle X is a circumscribed angle of circle V. Which name best describes figure VWXY?
square
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Points A, B, C, and D lie on circle M. Line segment BD is a diameter. What is the measure of angle ACD?
67.5
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What is the measure of circumscribed ∠X?
90
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What is the measure of angle COA?
160
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Points N, P, and R all lie on circle O. Arc PR measures 120°. How does the measure of angle RNQ relate to the measure of arc PR?
Angle RNQ is equal in measure to arc PR.
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Which point is on the circle centered at the origin with a radius of 5 units? Distance formula: (x2 - x1)2 + (y2-y1)2
A. 2, 21
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Mrs. Culland is finding the center of a circle whose equation is x2 + y2 + 6x + 4y - 3 = 0 by completing the square. Her work is shown. x2 + y2 + 6x + 4y - 3 = 0 x2 + 6x + y2 + 4y - 3 = 0 (x2 + 6x) + (y2 + 4y) = 3 (x2 + 6x + 9) + (y2 + 4y + 4) = 3 + 9 + 4
(x + 3)2 + (y + 2)2 = 42, so the center is (-3, -2).
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What is the radius of a circle whose equation is x2 + y2 + 8x - 6y + 21 = 0?
2
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Which equation represents a circle with a center at (-4, 9) and a diameter of 10 units?
(x + 4)2 + (y - 9)2 = 25
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Which equation represents a circle with a center at (-3, -5) and a radius of 6 units?
(x + 3)2 + (y + 5)2 = 36
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The center of a circle represented by the equation (x + 9)2 + (y − 6)2 = 102 is .
(-9,6)
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In the diagram, a circle centered at the origin, a right triangle, and the Pythagorean theorem are used to derive the equation of a circle, x2 + y2 = r2. If the center of the circle were moved from the origin to the point (h, k) and point P at (x, y) remains on the edge of the circle, which could represent the equation of the new circle?
(x - h)2 + (y - k)2 = r2
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Which explains how to find the radius of a circle whose equation is in the form x2 + y2 = z?
The radius is the square root of the constant term, z.
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What is the center of a circle whose equation is x2 + y2 + 4x - 8y + 11 = 0?
(-2, 4)
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Does the point (2,6 ) lie on the circle shown? Explain.
D. No, the distance from (0, 0) to (2,6 ) is not 3 units.
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Which graph represents the equation y2 = -4x?
A
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A parabola has a vertex at (0,0). The focus of the parabola is located on the positive x-axis. Which part of the graph will the directrix pass through?
the negative part of the x-axis
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A parabola has a vertex at (0,0). The equation for the directrix of the parabola is x = -4. In which direction does the parabola open?
right
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A parabola can be represented by the equation y2 = -x. What are the coordinates of the focus and the equation of the directrix
A
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A parabola can be represented by the equation x2 = -20y. What are the coordinates of the focus of the parabola?
(0,-5)
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The focus of a parabola is located at (0,-2). The directrix of the parabola is represented by y = 2. Which equation represents the parabola?
x2 = -8y
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A parabola has a vertex at (0,0). The focus of the parabola is located on the positive y-axis. In which direction must the parabola open?
up
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Which equation represents the parabola shown on the graph?
x2 = 6y
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A parabola, with its vertex at (0,0), has a focus on the negative part of the y-axis. Which statements about the parabola are true? Check all that apply.
The directrix will cross through the positive part of the y-axis. The equation of the parabola could be x2 = -1/2 y.
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A parabola is represented by the equation x2 = 4y. What are the coordinates of the focus of the parabola?
(0,1)
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If line segment AB measures approximately 8.6 units and is considered the base of parallelogram ABCD, what is the approximate corresponding height of the parallelogram? Round to the nearest tenth.
4.8 units
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What is the area of triangle QRS?
7
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What is the area of triangle GHJ?
6 square units
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If line segment BC is considered the base of triangle ABC, what is the corresponding height of the triangle?
1.6 units
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What is the area of parallelogram ABCD?
13
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Which expression can be used to find the area of triangle RST?
(8 ∙ 4) - 1/2 (10 + 12 + 16)
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What is the area of triangle ABC?
3
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What is the area of triangle LMN?
4 square units
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What is the area of parallelogram ABCD?
20 square units
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Michael is finding the area of parallelogram ABCD. To do this, he follows the steps in the table. To solve the problem using these steps, what are the dimensions of the rectangle he should draw?
4 units by 4 units
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How do the areas of the parallelograms compare?
The area of parallelogram ABCD is equal to the area of parallelogram EFGH.
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If line segment RU is considered the base of parallelogram RSTU, what is the corresponding height of the parallelogram?
5.4 units
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Payton cut out two shapes, as shown, that she will later put together to resemble a house. What is the total area of the two shapes
21 square units
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What is the area of triangle ABC?
7
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A hole the size of a photograph is cut from a red piece of paper to use in a picture frame. What is the area of the piece of red paper after the hole for the photograph has been cut?
47 square units
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A kite has vertices at (2, 4), (5, 4), (5, 1), and (0, -1). What is the approximate perimeter of the kite? Round to the nearest tenth.
16.8 units
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A company designs a logo using a kite figure around the letter t. The logo is 12 centimeters wide and 16 centimeters tall. What is the area of the logo?
96 sq. cm
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The area of the trapezoid is 40 square units. What is the height of the trapezoid?
5
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Trapezoid ABCD is graphed in a coordinate plane. What is the area of the trapezoid?
24 square units
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Kite WXYZ is graphed on a coordinate plane. What is the approximate perimeter of the kite? Round to the nearest tenth.
16.2 units
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What is the area of the rhombus?
24
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A kite is inscribed in a square with a side length of 9 units. What is the area of the kite?
40.5 square units
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The area of rhombus ABCD is 72 square units. EC = 8 units and DB = x - 1. What is the value of x?
10
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Figure ABCD is a kite. The area of ABCD is 48 square units. The length of line segment BD is 8 units. What is the length of AC?
12 units
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In the kite, AC = 10 and BD = 6. What is the area of kite ABCD?
30 square units
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Which statements are true about the regular polygon? Check all that apply.
Each interior angle measures 108°. All of the angles are congruent. The sum of the measures of the interior angles is 180(5 - 2)°.
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Which is a correct description of the polygon?
It is a convex pentagon because it has five sides and none of the sides would extend into the inside of the polygon.
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What is the value of x?
8
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What type of polygon is shown?
convex hexagon
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What type of polygon is shown?
concave heptagon
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What is the value of x?
130°
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Which polygon has an interior angle sum of 1080°?