Geometry AIR Test Mar 14-3:07 PM Congruence and Proof 33-39% coordinate/algebraic proofs, parallel and perpendicular lines, distance formula, midpoint formula. missing sides on triangles (trig ratios, Pythagorean Theorem, Law of Sines, Law of Circles 18-24% Probability 18-22% *Modeling and Reasoning will be tested a minimum of 20% throughout all these categories. Mar 14-3:09 PM 1
Formula Sheet formula sheet! Mar 15-8:11 AM 2 2 Mar 15-8:14 AM 2
Mar 15-8:21 AM Some questions will have you plot points, graph, fill-in-the-blank, and select multiple answers! Read the questions carefully! Mar 15-8:21 AM 3
Fraction: division 4/5 4 5 Decimal to fraction: 1: Frac ENTER ENTER Graphing: y =, use XTθn for variable to input equation 2nd to access table 2nd TRACE to access calculations menu, where you can do things such as... 2: zero 3: minimum 4: maximum 5: intersect MODE to quit. Mar 15-8:24 AM Mar 15-8:24 AM 4
Quadrilateral BCDE is shown on the coordinate grid. Keisha reflects the figure across the line y = x to create B'C'D'E'. y = x x y 1 1 2 2 3 3 Use the Connect Line tool to draw quadrilateral B'C'D'E'. B(3, -3) (-3, 3) C(9, -4) (-4, 9) D(10,-10) (-10, 10) E(2, -5) (-5, 2) because y = -x 9
Hash marks to show which lines "match" Prove: BAC BCA Place reasons in the table to complete the proof. Statements Reasons 1. Triangle ABC is isosceles. D is the midpoint of AC. 1. Given 2. AD DC 2. Definition of midpoint. 3. BA BC 3. Definition of isosceles triangle 4. A single line segment can 4. BD exists. be drawn between any two points. 5. BD BD 5. Reflexive property 6. ABD CBD 6. SSS Congruency 7. BAC BCA 7. CPCTC General Process for Proofs: 1. Mark any given information in picture (if no picture, might be helfpul to make one!) 2. Follow proof; mark additional information in proof to picture 3. Read through proof line by line. What are they doing? How did they get each line? *know vocabulary and properties--these will provide you with most of your answers! *need hints? Look at your answer options--the right answer is there somewhere! #6 Use picture to see that SSS is what is needed. #7 Because triangles are congruent, the matching parts are too! CPCTC is a very common reason to a proof. 10
The proof shows that opposite angles of a parallelogram are congrent. Prove: BAD DCB Proof: Statements Reasons ABCD is a parallelogram with diagonal AC. AB ll CD and AD ll BC Given Definition of parallelogram The only between lines is the replacement of angle 2 with angle 3. 2 3 1 4 m 2 = m 3 and m 1 = m 4 Alternate interior angles are congruent Measures of congruent angles are equal m 1 + m 2 = m 4 + m 2 Addition property of equality m 1 + m 2 = m 4 + m 3? m 1 + m 2 = m BAD m 3 + m 4 = m DCB Angle addition postulate m BAD = m DCB Substitution BAD DCB Angles are congruent when their measures are equal What is the missing reason in this partial proof? b. Substitution Because we were already told angle 2 is congruent to angle 3. 11
Three vertices of parallelogram PQRS are shown: Q (8,5), R (5, 1), S (2, 5) May be helpful to graph! Place statements and reasons in the table to complete the proof that shows that parallelogram PQRS is a rhombus. Statements SR = 5 Reasons Pythagorean Theorem QR = 5 SR = QR SR QR PS QR Substitution Definition of congruent line segments Property of a parallelogram SR PQ Property of a parallelogram Parallelogram PQRS is a rhombus. Definition of a rhombus PQ = 5 SR = 7 PQ = 7 QR = 7 PSR = 90 Definition of perpendicular Definition of parallel lines lines d (SR)= (x - x) 2 +(y-y) 2 = (5-2) 2 +(1-5) 2 = (3) 2 +(-4) 2 = 9+16 = 25 = 5 sides are equal, so we know we need the length of the lines (your options for lengths are 5 and 7). use the distance formula for QR and SR, so that should be the first two lines in the proof. The last bit that we need simply is using that if we have one side as 5, then we know the opposite side is 5 because that is how a parallelogram works. This was also very close to the line above it in the proof. 12
Angle RPQ is inscribed, arc 2 Angle ROQ is central, equals arc Both angles open to the same arc on the circle. The teacher asks students to select the correct claim about the relationship between m RPQ and m ROQ. No, because they have different relationships with the arc. > Claim 1: The measure of RPQ is equal to the measure of ROQ. > Claim 2: The measure of ROQ is twice the measure of RPQ. YES! Claim 2 is correct. Since angle RPQ is inscribed, its relationship with the intercepted arc is arc 2. In arc with angle ROQ, but since angle ROQ is central angle ROQ is equal to the arc. 13
capsule like the one shown. The capsule has a radius of 3 millimeters (mm) and a length of 15 mm. V(sphere)=4/3* r 3 r = 3 (can rule out V(cylinder) = r 2 *h c and d) h of the entire capsule is 15, but that includes the radius of both hemispheres, so Which statement best explains how to find the amount of vitamin mix that fits in the capsule? with a radius of 3 millimeters and a height of 9 millimeters. with a radius of 3 millimeters and a height of 15 millimeters. with a radius of 6 millimeters and a height of 9 millimeters. with a radius of 6 millimeters and a height of 15 millimeters. 14
through point P? (looks to be a complete construction!) In the running, but the only thing This shows the correct first step. Need to mark equal distance around P so that you can construct perpendicular likely not the first step. 15
d(bc) = A (3 - -1) 2 + (-1-6) 2 x y x y x y = (4) 2 + (-7) = 9 + 36 Perimeter = total distance around the outside d = (x 2 - x 1 ) 2 2 - y 1 ) 2 d(ab) = (-1 - -4) 2 + (6-0) 2 = (3) 2 + (6) 2 = 9 + 36 = 45 d(bc) = (3 - -1) 2 + (-1-6) 2 = (4) 2 + (-7) 2 = 16 + 49 = 65 d(ab) = (3 - -4) 2 + (-1-0) 2 = (7) 2 + (-1) 2 = 49 + 1 = 50 16
Allison designs fancy boxes to fill with chocolates. The boxes are in the shape of a right square pyramid as shown, where 8y represents the length of one side of the base of the pyramid, and 5y represents the height of one triangular face of the pyramid. For height of pyramid, look May be useful to draw picture: at triangle inside. Bottom? 4y as it is half the length of 5y the base.? 4y 8y 8y (shown below) to determine the height is volume formula. Need height of pyramid for volume formula! 1000 = 1/3*8y*8y*3y = 64y 3 The large size box must be designed to have a volume of 1,000 cubic centimeters. A. Create an equation that can be used to calculate the length of the base and height of the triangular face of the box. Enter your equation in the first response box. B. What dimensions for the length, in centimeters, of the base and height of the triangular face, in centimeters, satisfy these constraints? 1000=64y 3 B. Length of Base = 20 centimeters 12.5 (2.5*8) (2.5*5) Pythagorean Theorem for height of pyramid: a 2 + b 2 = c 2 a 2 + (4y) 2 = (5y) 2 a 2 + 16y 2 = 25y 2-16y 2-16y 2 a 2 = 9y 2 a 2 = 9y 2 a = 3y 1000 = 64y 3 64 64 15.625 = y 3 15.625 = y 3 2.5 = y 17
Kyle performs a transformation on a triangle. The resulting triangle is similar but not congruent to the original triangle. Which transformation did Kyle perform on the triangle? a. dilation size b. reflection c. rotation d. translation do not change size 18
Triangle PQR is shown, where ST is parallel to RQ. SR TQ Marta wants to prove that =. PS PT Place a statement or reason in each blank box to complete Marta's proof. Statements Reasons 1. ST RQ 1. Given 2. PST R and PTS Q 2. If two parallel lines are cut by a transversal, then corresponding angles are congruent. 3. PQR ~ PTS AA Similarity PR = PQ PS PT Corresponding sides of similar triangles are proportional. 5. PR = PS + SR, PQ = PT + TQ 6. PS + SR = PT + TQ PS PT 7. PS + SR = PT + TQ PS PS PT PT 8. SR = TQ PS PT 5. Segment addition postulate 6. Substitution 7. Commutative property of addition. 8. Subtraction property of P equality P Line 3: Look at picture--two angles marked, AA. Line 4: Hints are in lines 5 and 6. How did they get those fractions? Need a fraction here, make sure you pick the correct relationship from the answer options (corresponding sides). 19
A total of 200 people attend a party, as shown in the table. A person is selected at random to win a prize. The probability of selecting a female is 0.6. The probability of selecting a child, given that the person is female, is 0.25. The probability of selecting a male, given that the person is a child, is 0.4. A: child given B: female P(B) = 0.6, P(A l B) = 0.25 Complete the two-way table to show the number of adults, children, males, and females who attended the party. 60 20 90 30 P(A l B) = P(A + B) P(B) 0.25 = P(A + B) 0.6 *0.6 *0.6 0.15 = P(A + B) 200*0.15 = 30 (child + female) Use table to calculate other values after 30.? + 30 = 50 children, so?=20? + 30 = 120 females, so?=90 Then use to get Adult males. 20
A translation is applied to PQR to create P'Q'R'. y positive + left 4, -4 x is negative - up 3, +3 x is positive + y is negative - Let the statement (x, y) (a, b) describe the translation. Create equations for a in terms of x and for b in terms of y that could be used to describe the translation. Up 3 y + 3 Left 4 x - 4 } a = x - 4, b = y + 3 21