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anonymous

  • one year ago

If Triangle ABC ~ triangle DEC solve for x. Picture not drawn to scale. a. x=2 b. x=3 c. x=3 d. x=13

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  1. anonymous
    • one year ago
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  2. mathstudent55
    • one year ago
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    First, you need to find which sides are corresponding sides. |dw:1437336453516:dw| Complete the sentences below: Side 1 corresponds to side Side 2 corresponds to side Side 3 corresponds to side

  3. anonymous
    • one year ago
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    side 1 corresponds to side 4 side 2 corresponds to side 5 side 3 corresponds to side 6 i think

  4. mathstudent55
    • one year ago
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    This is a little tricky, that why I asked. 1 corresponds to 4. That is correct. now let's look at the other sides. Since sides 1 and 4 are parallel, we can conclude some pairs of angles are congruent by interior angles of parallel lines cut by a tranversal.

  5. mathstudent55
    • one year ago
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    |dw:1437337115089:dw|

  6. mathstudent55
    • one year ago
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    You understand how the pairs of corresponding angles are congruent?

  7. anonymous
    • one year ago
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    no

  8. mathstudent55
    • one year ago
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    Ok. I'll backtrack.

  9. mathstudent55
    • one year ago
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    Let's start with 2 parallel lines. Lines m and n are given as parallel. |dw:1437337278594:dw|

  10. mathstudent55
    • one year ago
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    Now we draw a transversal. Line t is a transversal to lines m and n. Angles 1 and 2 are alternate interior angles. When parallel lines are cut by a transversal, alternate interior angles are parallel. That means angles 1 and 2 are congruent. |dw:1437337364835:dw|

  11. mathstudent55
    • one year ago
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    Ok so far?

  12. anonymous
    • one year ago
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    yes :)

  13. mathstudent55
    • one year ago
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    Now we draw another tranversal, line u. We find 2 alternate interior angles, labeled 3 and 4, and we know they must be congruent. |dw:1437337588277:dw|

  14. mathstudent55
    • one year ago
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    Notice, we have the same drawing as your problem. The other two angles of the triangles are also congruent because they are vertical angles. |dw:1437337682486:dw|

  15. mathstudent55
    • one year ago
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    As you can see, our two triangles are similar. There are 3 pairs of congruent corresponding angles.

  16. mathstudent55
    • one year ago
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    Now let's get back to our problem. We need to figure out which pairs of sides are corresponding.

  17. mathstudent55
    • one year ago
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    |dw:1437337781397:dw|

  18. mathstudent55
    • one year ago
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    The three angles of the two triangles are marked with 1 line, 2 lines or 3 lines. Each side included by corresponding angles is corresponding. Look at side 1 on the left. Its angles have 1 mark and 2 marks. Then look at side 4. Its angles also have 1 mark and 2 marks. Side 1 and side 4 are corresponding. |dw:1437337945253:dw|

  19. mathstudent55
    • one year ago
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    Sides 2 and 6 have 2 marks and 3 marks on the angles at their ends, so sides 2 and 6 are corresponding sides. See figure below. |dw:1437337994043:dw|

  20. mathstudent55
    • one year ago
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    Finally, we do the same for sides 3 and 5. Side 3 and 5 are corresponding. |dw:1437338122927:dw|

  21. anonymous
    • one year ago
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    oh ok i understand that now

  22. mathstudent55
    • one year ago
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    Now that we know which sides are corresponding sides, we can write a proportion of side lengths. \(\dfrac{AC}{DC} = \dfrac{BC}{EC} \) \(\dfrac{6}{8 + x} = \dfrac{11-x}{19 - 2x} \) Ok so far?

  23. anonymous
    • one year ago
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    yes :)

  24. mathstudent55
    • one year ago
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    We solve the equation by cross multiplying. \(6(19 - 2x) = (8 + x)(11 - x) \) \(114 - 12x = 88 - 8x + 11x - x^2\)

  25. mathstudent55
    • one year ago
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    \(x^2 - 15 x + 26 = 0\) \((x - 13)(x - 2) = 0\) \(x = 13\) or \(x = 2\)

  26. mathstudent55
    • one year ago
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    Now let's look at all sides and plug in 2 and 13 for x and calculate all lengths: Side x = 2 x = 13 6 6 6 8 + x 10 21 11 - x 9 -2 19 - 2x 15 -7 Notice that when we use x = 13, some sides end up with a negative length. Since the length of the side of a triangle must be positive, x = 13 is eliminated, and the solution is x = 2

  27. anonymous
    • one year ago
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    oh i understand it now, thank you for all the help!

  28. mathstudent55
    • one year ago
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    You're welcome.

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