## TommyTrojan 2 years ago Nick plotted A(2, 2), B(3, 4), and C(4, 2) and joined the points to form triangle ABC. He plotted two other points at P(-3, 4) and Q(-2, 2). What should be the coordinates of the third point R to form triangle PQR that is congruent to triangle ABC? (-1, 4) (-2, -5) (-5, 2) (-4, 1)

1. TommyTrojan

2. AccessDenied

Have you tried plotting the points that you were given?

3. TommyTrojan

Yes, but I still don't get it

4. AccessDenied

Could you post that graph here? :)

5. TommyTrojan

I actually wrote it on loose leaf paper lol. Soz

6. TommyTrojan

But how would you do it?

7. AccessDenied

My first step is graphing the points. Then, I'd look for any obvious solutions like if there were symmetries between the points...

8. TommyTrojan

OK can I ask you a few more?

9. TommyTrojan

Its either a or b

10. AccessDenied

Oh, I just realised something -- the points have to correspond for the congruence! It's been a bit since I've done Geometry... lol. :P Let me graph it myself quick.

11. TommyTrojan

OK thank you lol no problem

12. AccessDenied

PQ = AB QR = BC PR = AC

13. TommyTrojan

So it is either a or b

14. AccessDenied

We could either do something like guess and check each point to see if it is the same, or maybe use some algebra and find the point which is a distance of BC from Q and a distance of AC from P

15. TommyTrojan

I got a. Is that right?

16. AccessDenied

A) appears correct to me as well. :) For Geometry, if it weren't so obvious, we could use distance formula for some arbitrary point R(x,y) that meets those two distance conditions and solve that two-variable system. This one is pretty simple to "see" though.

17. TommyTrojan

I have a few more questions, if that's all right with you?

18. AccessDenied

I think we'd get two solutions, one would be some sort of fraction or rational and (-1,4) also.

19. AccessDenied

Sure, that's fine. :)

20. TommyTrojan

OK thank you so much Triangle PQR is similar to triangle ABC in the figure below. What is the perimeter of triangle ABC? 10.5 inches 39.9 inches 42.0 inches 60.65 inches

21. TommyTrojan

22. TommyTrojan

I got B, just checking my work.

23. TommyTrojan

@AccessDenied

24. AccessDenied

How did you get B)?

25. TommyTrojan

Tell you the truth, I guessed, i'm confused on how to do it.

26. AccessDenied

Ah, okay. Well, we'd like to first find that one side length that we don't know on the triangle. Ratios of corresponding sides on similar triangles have to be the same, so this is one way to approach it. Set up this proportion of the sides: $$\displaystyle \frac{\text{AC}}{\text{PR}} = \frac{\text{AB}}{\text{PQ}}$$ Notice how these are in fact corresponding sides; we can tell by which angles they are between, or just by position in the name of the triangles. $$\color{green}{\textbf{A}}\text{B}\color{green}{\textbf{C}} ~ \color{green}{\textbf{P}}\text{Q}\color{green}{\textbf{R}}$$

27. AccessDenied

So, can you find the length of that side (AC) given this information?

28. TommyTrojan

Yes I got 10.5.

29. AccessDenied

Correct. :) Now, we just sum the lengths of the sides.

30. TommyTrojan

Your VERY helpful, but I have alot more questions if that's all right with you. PS i'll medal you when we're all done, or now if you want.

31. AccessDenied

Hmm, how many questions? :)

32. TommyTrojan

5 lol

33. TommyTrojan

Most of them are just checking though

34. AccessDenied

Okay. :)

35. TommyTrojan

Ok let's get started then lol. Jeremiah uses bamboo rods to make the frame of a tailless kite. He ties three bamboo rods together to form a right triangle PQR. He then ties another rod from P that meets RQ at a right angle. Segment PS in the figure below represents this rod and it is 4 inches long. Which of the following could be the lengths of segments QS and SR? QS = 2 inches, SR = 8 inches QS = 6 inches, SR = 10 inches QS = 2 inches, SR = 2 inches QS = 4 inches, SR = 12 inches

36. TommyTrojan

37. TommyTrojan

I got A. Is that correct?

38. AccessDenied

If I recall this type of situation correctly: PS/ QS = RS/ PS 4 / QS = RS / 4 A) 4/2 = 8/4 ==> 2 = 2. Appears to be correct. :)

39. TommyTrojan

OK An architect planned to construct two similar stone pyramid structures in a park. The figure below shows the front view of the pyramids in her plan but there is an error in the dimensions. Which of the following changes should she make to the dimensions to correct her error? change the length of side AB to 2 feet change the length of side PQ to 8 feet change the length of side AB to 1 feet change the length of side PQ to 4 feet

40. TommyTrojan

41. AccessDenied

Hmm, do you have an answer to check?

42. TommyTrojan

No this one I don't quite understand.

43. AccessDenied

Ah, okay. This is another similar figures question. The key here is having the same side ratios again, similar to some earlier problems. First, I'd check each side ratio to see if there is a sort of "odd-man out."

44. AccessDenied

Our corresponding sides: AC ~ PR, AB ~ PQ, BC ~ QR We should check all three of these ratios. AC/PR, AB/PQ, BC/QR

45. TommyTrojan

Ok i plugged in the numbers but now what?

46. AccessDenied

What do you get for each ratio? Is one of them different?

47. TommyTrojan

AB-PQ is the different on it gets 1/2, while all the other ratios get .3 repeated

48. AccessDenied

Yep. So, we'd like to find the solution that changes AB or PQ to match the other two side ratios.

49. AccessDenied

For this, we could just go through each proposed solution and check to see if it works out.

50. TommyTrojan

Is it A?

51. AccessDenied

Yes. :)

52. TommyTrojan

Ok only 2 more left.

53. TommyTrojan

Joshua used two wood beams, PC and QA, to support the roof of a model house. The beams intersect each other to form two similar triangles QRP and ARC as shown in the figure below. The length of segment PR is 1.8 inches and the length of segment CR is 3.3inches. The distance between A and C is 6.6 inches. What is the distance between the endpoints of the beams P and Q? 3.6 inches 0.9 inches 1.8 inches 2.5 inches

54. TommyTrojan

55. AccessDenied

Here is another case of similar triangles. Can you identify the side-ratios here? Just to see if you understand the similar triangles have side ratios detail. :)

56. TommyTrojan

Oh I forgot to trell you I got A.

57. AccessDenied

Ah, okay. A) is correct to me. :) We set up a simple side ratio: QR/PR = AC/PQ' 3.3/1.8 = 6.6/x; x = 6.6 * 1.8 / 3.3 = 3.6

58. TommyTrojan

Moris drew two triangles; triangle ABC and triangle PQR, on a coordinate grid. The coordinates of the vertices of triangle PQR are P(-3, -2), Q(-3, -4), and R(-1, -4). The coordinates of the vertices of triangle ABC are A(-3, 4), B(-1, 4), C(-3, 2). Which postulate can be used to prove that the two triangles are congruent? SAS, because AAA, because ASA, because SSS, because

59. TommyTrojan

I got D

60. AccessDenied

Yeah, I am thinking D) as well. We're given all the points, we could easily just take distance formula or if they're horixontal/vertical, count.

61. TommyTrojan

Well I guess that's it, I have more but, since I only said 5 I will keep my word and let you go. OK Thanks though

62. TommyTrojan

Thanks BIG TIME!!

63. AccessDenied

You're welcome! :)