February16 February20

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Problems and Solutions for the week of Feb 16 - Feb 20

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February16 February20

  1. 1. Math Tutorial Questions For the week of February 16-20
  2. 2. Identifying Types of Lines and Planes Questions (3.1.1) – February 16, 2009 <ul><li>Use the diagram below to identify each of the following. </li></ul><ul><ul><li>A pair of parallel segments </li></ul></ul><ul><ul><li>A pair of skew segments </li></ul></ul><ul><ul><li>A pair of perpendicular segments </li></ul></ul><ul><ul><li>A pair of parallel planes </li></ul></ul><ul><li>Use the diagram to the right to identify each of the following. </li></ul><ul><ul><li>A pair of parallel segments </li></ul></ul><ul><ul><li>A pair of skew segments </li></ul></ul><ul><ul><li>A pair of perpendicular segments </li></ul></ul><ul><ul><li>A pair of parallel planes </li></ul></ul>
  3. 3. Identifying Types of Lines and Planes Solutions (3.1.1) – February 16, 2009 <ul><li>Use the diagram to the right to identify each of the following. </li></ul><ul><ul><li>3 example solutions are listed, there are other possibilities </li></ul></ul><ul><ul><li>A pair of parallel segments </li></ul></ul><ul><ul><li>AB & CD, CB & GF, EF & HG </li></ul></ul><ul><ul><li>A pair of skew segments </li></ul></ul><ul><ul><li>AE & GF, AB & DH, HG & BF </li></ul></ul><ul><ul><li>A pair of perpendicular segments </li></ul></ul><ul><ul><li>DH & HG, AD & DC, EF & FG </li></ul></ul><ul><ul><li>A pair of parallel planes </li></ul></ul><ul><ul><li>Plane DCG & Plane ABF, Plane ADH & Plane BCG, Plane ABC & Plane EFG </li></ul></ul><ul><li>Use the diagram to the right to identify each of the following. </li></ul><ul><ul><li>3 example solutions are listed, there are other possibilities </li></ul></ul><ul><ul><li>A pair of parallel segments </li></ul></ul><ul><ul><li>KL & NM, LQ & MR, QR & PS </li></ul></ul><ul><ul><li>A pair of skew segments </li></ul></ul><ul><ul><li>KL & MR, LQ & PS, PQ & NS </li></ul></ul><ul><ul><li>A pair of perpendicular segments </li></ul></ul><ul><ul><li>KL & LQ, LQ & QR, KN & NS </li></ul></ul><ul><ul><li>A pair of parallel planes </li></ul></ul><ul><ul><li>Plane KLM & Plane PQR, Plane MLQ & Plane NKP, Plane NMR & Plane KLQ </li></ul></ul>
  4. 4. Classifying Pairs of Angles <ul><li>Use the diagram to the right to identify each of the following. </li></ul><ul><ul><li>A pair alternate interior angles </li></ul></ul><ul><ul><li>A pair of corresponding angles </li></ul></ul><ul><ul><li>A pair of alternate exterior angles </li></ul></ul><ul><ul><li>A pair of same-side interior angles </li></ul></ul>Questions (3.1.2) – February 17, 2009 <ul><li>Use the diagram to the right to identify each of the following. </li></ul><ul><ul><li>A pair alternate interior angles </li></ul></ul><ul><ul><li>A pair of corresponding angles </li></ul></ul><ul><ul><li>A pair of alternate exterior angles </li></ul></ul><ul><ul><li>A pair of same-side interior angles </li></ul></ul>
  5. 5. Classifying Pairs of Angles <ul><li>Use the diagram to the right to identify each of the following. </li></ul><ul><li> 2 example solutions are listed, there are other possibilities </li></ul><ul><ul><li>A pair alternate interior angles </li></ul></ul><ul><ul><li>Angles 3 & 5, Angles 4 & 6 </li></ul></ul><ul><ul><li>A pair of corresponding angles </li></ul></ul><ul><ul><li>Angles 2 & 6, Angles 4 & 8 </li></ul></ul><ul><ul><li>A pair of alternate exterior angles </li></ul></ul><ul><ul><li>Angles 2 & 8, Angles 1 & 7 </li></ul></ul><ul><ul><li>A pair of same-side interior angles </li></ul></ul><ul><ul><li>Angles 4 & 5, Angles 3 & 6 </li></ul></ul>Solutions (3.1.2) – February 17, 2009 <ul><li>Use the diagram to the right to identify each of the following. </li></ul><ul><li>1 example solution is listed, there are other possibilities </li></ul><ul><ul><li>A pair alternate interior angles </li></ul></ul><ul><ul><li>Angle EHG & Angle HGK </li></ul></ul><ul><ul><li>A pair of corresponding angles </li></ul></ul><ul><ul><li>Angle EHG & Angle FGJ </li></ul></ul><ul><ul><li>A pair of alternate exterior angles </li></ul></ul><ul><ul><li>Angle IHE & Angle JGK </li></ul></ul><ul><ul><li>A pair of same-side interior angles </li></ul></ul><ul><ul><li>Angle EHG & Angle FGH </li></ul></ul>
  6. 6. Angles Formed by Parallel Lines & Transversals Questions (3.2.3) – February 18, 2009 <ul><li>Use the diagram below to find each angle measure. </li></ul><ul><ul><li>m ECF </li></ul></ul><ul><ul><li>m DCE </li></ul></ul><ul><li>Find x and y in the diagram below. </li></ul>
  7. 7. Angles Formed by Parallel Lines & Transversals Solutions (3.2.3) – February 18, 2009 <ul><li>Use the diagram below to find each angle measure. </li></ul><ul><ul><li>Corresponding angles are equal in measure. </li></ul></ul><ul><ul><li>Angle ECF & Angle EBG are corresponding. </li></ul></ul><ul><ul><li>Angle DCE & Angle ABE are corresponding. </li></ul></ul><ul><ul><li>m ECF </li></ul></ul><ul><ul><li>Angle ECF = Angle EBG (Substitute values in) </li></ul></ul><ul><ul><li>Angle ECF = 70º </li></ul></ul><ul><ul><li>m DCE </li></ul></ul><ul><ul><li>Angle DCE = Angle ABE (Substitute values in) </li></ul></ul><ul><ul><li>5x = 4x + 22 (Subtract 4x from both sides) </li></ul></ul><ul><ul><li> x = 22 </li></ul></ul><ul><li>Find x and y in the diagram below. </li></ul><ul><li>Corresponding angles are equal in measure. </li></ul><ul><li>5x + 5y = 60 (Subtract 5x from both sides) </li></ul><ul><li> 5y = 60 – 5x (Divide both sides by 5) </li></ul><ul><li> y = 12 – x </li></ul><ul><li>Alternate Interior angles are equal in measure. </li></ul><ul><li>5x + 4y = 55 (Substitute y = 12 – x in for y) </li></ul><ul><li>5x + 4(12 – x) = 55 (Multiply 4 through 12 – x) </li></ul><ul><li>5x + 48 – 4x = 55 (Collect x terms together) </li></ul><ul><li>x + 48 = 55 (Subtract 48 from both sides) </li></ul><ul><li>x = 7 </li></ul>y = 12 – x (Substitute 7 in for x) y = 12 – 7 (Subtract 12 and 7) y = 5
  8. 8. Slopes of Lines Questions (3.5.2) – February 19, 2009 <ul><li>Use the diagram below and the information above to determine the slope of each line. </li></ul><ul><ul><li>AB </li></ul></ul><ul><ul><li>AC </li></ul></ul><ul><ul><li>AD </li></ul></ul><ul><ul><li>CD </li></ul></ul><ul><li>Justin is driving from home to his college dormitory. At 4:00 P.M., he is 260 miles from home. At 7:00 P.M., he is 455 miles from home. Use the graph of the line that represents Justin’s distance from home at a given time. Find and interpret the slope of the line. </li></ul>
  9. 9. Slopes of Lines Solutions (3.5.2) – February 19, 2009 <ul><li>Use the diagram to the right to determine the slope of each line. </li></ul><ul><ul><li>A(-2, 7), B(3, 7), C(4, 2), D(-2, 1) </li></ul></ul><ul><ul><li>AB </li></ul></ul><ul><ul><li>Line AB is horizontal, therefore the slope is 0. </li></ul></ul><ul><ul><li>AC </li></ul></ul><ul><ul><li>AD </li></ul></ul><ul><ul><li>Line AD is vertical, therefore the slope is UNDEFINED. </li></ul></ul><ul><ul><li>CD </li></ul></ul><ul><li>Justin is driving from home to his college dormitory. At 4:00 P.M., he is 260 miles from home. At 7:00 P.M., he is 455 miles from home. Use the graph of the line that represents Justin’s distance from home at a given time. Find and interpret the slope of the line. </li></ul>The slope is 65 mi/hr, which means his average speed while driving home was at a rate of 65 miles per hour.
  10. 10. Using Slopes to Classify Pairs of Lines Questions (3.5.3) – February 20, 2009 <ul><li>Graph each pair of lines. Find their slopes and use them to determine whether the lines are parallel, perpendicular, or neither. </li></ul><ul><ul><li>UV and XY for U(0, 2), </li></ul></ul><ul><ul><li>V(-1, -1), X(3, 1), Y(-3, 3) </li></ul></ul><ul><ul><li>GH and IJ for G(-3, -2), </li></ul></ul><ul><ul><li>H(1, 2), I(-2, 4), J(2, -4) </li></ul></ul><ul><ul><li>CD and EF for C(-1, -3), </li></ul></ul><ul><ul><li>D(1, 1), E(-1, 1), F(0, 3) </li></ul></ul>
  11. 11. Using Slopes to Classify Pairs of Lines Solutions (3.5.3) – February 20, 2009 <ul><ul><li>UV and XY </li></ul></ul><ul><ul><li>U(0, 2) </li></ul></ul><ul><ul><li>V(-1, -1) </li></ul></ul><ul><ul><li>X(3, 1) </li></ul></ul><ul><ul><li>Y(-3, 3) </li></ul></ul><ul><ul><li>GH and IJ </li></ul></ul><ul><ul><li>G(-3, -2) </li></ul></ul><ul><ul><li>H(1, 2) </li></ul></ul><ul><ul><li>I(-2, 4) </li></ul></ul><ul><ul><li>J(2, -4) </li></ul></ul><ul><li>Graph each pair of lines. Find their slopes and use them to determine whether the lines are parallel, perpendicular, or neither. </li></ul><ul><ul><li>CD and EF </li></ul></ul><ul><ul><li>C(-1, -3) </li></ul></ul><ul><ul><li>D(1, 1) </li></ul></ul><ul><ul><li>E(-1, 1) </li></ul></ul><ul><ul><li>F(0, 3) </li></ul></ul>The slopes are 3 and -⅓, which multiply to equal -1, or are called opposite reciprocals of each other. Therefore the lines are perpendicular lines. The slopes are 1 and -2, which do not multiply to equal -1 and they are not the same slope. Therefore the lines are not perpendicular lines, and they are not parallel lines. The slopes are 2 and 2, which means they have the same slope. Therefore the lines are parallel lines.

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