anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
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mathstudent55
  • mathstudent55
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
anonymous
  • anonymous
side 1 corresponds to side 4 side 2 corresponds to side 5 side 3 corresponds to side 6 i think

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mathstudent55
  • mathstudent55
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.
mathstudent55
  • mathstudent55
|dw:1437337115089:dw|
mathstudent55
  • mathstudent55
You understand how the pairs of corresponding angles are congruent?
anonymous
  • anonymous
no
mathstudent55
  • mathstudent55
Ok. I'll backtrack.
mathstudent55
  • mathstudent55
Let's start with 2 parallel lines. Lines m and n are given as parallel. |dw:1437337278594:dw|
mathstudent55
  • mathstudent55
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|
mathstudent55
  • mathstudent55
Ok so far?
anonymous
  • anonymous
yes :)
mathstudent55
  • mathstudent55
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|
mathstudent55
  • mathstudent55
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|
mathstudent55
  • mathstudent55
As you can see, our two triangles are similar. There are 3 pairs of congruent corresponding angles.
mathstudent55
  • mathstudent55
Now let's get back to our problem. We need to figure out which pairs of sides are corresponding.
mathstudent55
  • mathstudent55
|dw:1437337781397:dw|
mathstudent55
  • mathstudent55
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|
mathstudent55
  • mathstudent55
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|
mathstudent55
  • mathstudent55
Finally, we do the same for sides 3 and 5. Side 3 and 5 are corresponding. |dw:1437338122927:dw|
anonymous
  • anonymous
oh ok i understand that now
mathstudent55
  • mathstudent55
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?
anonymous
  • anonymous
yes :)
mathstudent55
  • mathstudent55
We solve the equation by cross multiplying. \(6(19 - 2x) = (8 + x)(11 - x) \) \(114 - 12x = 88 - 8x + 11x - x^2\)
mathstudent55
  • mathstudent55
\(x^2 - 15 x + 26 = 0\) \((x - 13)(x - 2) = 0\) \(x = 13\) or \(x = 2\)
mathstudent55
  • mathstudent55
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
anonymous
  • anonymous
oh i understand it now, thank you for all the help!
mathstudent55
  • mathstudent55
You're welcome.

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