Discover Geometry Theorems for Simple Roof Truss Design | Figure 2.10

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Homework Help Overview

The discussion revolves around a geometry problem related to a simple roof truss design, specifically focusing on determining the lengths of segments XT and XZ based on given dimensions and relationships within the truss structure.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the use of the Pythagorean theorem to find lengths in right triangles formed by the truss structure. There is discussion about the relationships between segments and the need for additional information to resolve certain unknowns.

Discussion Status

Some participants have offered methods to set up equations based on the geometry of the problem, while others express confusion regarding the number of unknowns and the sufficiency of the provided information. The discussion is ongoing with various interpretations being explored.

Contextual Notes

There is mention of the original poster's request for guidance without complete solutions, indicating a focus on understanding rather than direct answers. Participants are grappling with the implications of missing information necessary for solving the problem fully.

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A simple roof truss design is shown in figure 2.10( i have attached) The lower section, VWXY is made from three equal length segments. UW and XZ are perpendicular to VT and TY, respectively. If VWXY is 20m and the height of the truss is 2.5m, determine the lengths of XT and XZ.

Ok It has been years since I have taken geometry and am looking for a little guidance. Any theorems or help to get this problem started in the right direction would be great. I do not want full solutions that would be defeating the purpose :)
 

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VWXY is 20m long and divided into 3 equal segments so each segment has length 20/3 m. If you draw a perpendicular from T to VWXY, it will bisect WX at, say S, and so forms a right triangle with legs of length (20/3)/2= 10/3 and 2.5 m. You can use the Pythagorean theorem to find the length of the hypotenuse XT: XT2= (10/3)2+ 2.52.

Similarly, that perpendicular forms right triangle STY having legs of length 2.5m and 10m (half of the 20m length of VWXY). TY2= 2.52+ 10[/sup]. Call that x.

Finding XZ is a little harder. Let u= length of TZ. Then length of ZY is x- u. You now have two right triangle with leg XZ. The first right triangle is XTZ which has hypotenuse
XT and legs TZ and XZ: TX2= XZ2+ u2. The other right triangle is XZY which has hypotenuse 10/3 m and legs XZ and TY- u: (10/3)2= XZ2+ (TY- u)2. Since you have calculated TX and TY above that gives two equations to solve for XZ and u.
 


Edit: Deleting my solutions, per OP request. *sorry*
 
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HallsofIvy said:
Finding XZ is a little harder. Let u= length of TZ. Then length of ZY is x- u. You now have two right triangle with leg XZ. The first right triangle is XTZ which has hypotenuse
XT and legs TZ and XZ: TX2= XZ2+ u2. The other right triangle is XZY which has hypotenuse 10/3 m and legs XZ and TY- u: (10/3)2= XZ2+ (TY- u)2. Since you have calculated TX and TY above that gives two equations to solve for XZ and u.

Ok maybe I am missing something. I can solve to get TY to be approximately 10.31. But after that I cannot see how to solve for TZ and ZY. There is always 2 things left unsolved and we all know you can't solve for 2 different things without at least one of them. I am a little confused. I understand the problem setup and where you are trying to go though just seems to be missing some important numbers to help solve.
 


You can, as I showed, use triangles XZT and XZY to set up two equations for those two unknown numbers.