Discover Geometry Theorems for Simple Roof Truss Design | Figure 2.10

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To eliminate u, multiply the first equation by (TY- u)2= (10.31- u)2 and the second by u2. That eliminates u from both equations. Now you have two equations in just one unknown, XZ. Solve one for u, substitute in the other and solve for XZ.In summary, we have a simple roof truss design with a lower section VWXY made of three equal length segments, each 20/3 m long. Using the Pythagorean theorem, we can determine the length of the hypotenuse XT and TY. To find XZ, we can set up two equations using triangles XZT and XZY and solve for XZ.
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ur5pointos2sl
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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.
 
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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.
 
  • #5


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

1. What are some common geometry theorems that I should know?

Some common geometry theorems include the Pythagorean Theorem, the Angle Sum Theorem, and the Triangle Inequality Theorem.

2. How can geometry theorems help me solve problems?

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Yes, geometry theorems have practical applications in fields such as architecture, engineering, and design. They can also be used in everyday scenarios, such as measuring angles for a home renovation project or calculating distances for a road trip.

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To prove a statement using geometry theorems, you must provide logical reasoning and evidence based on the theorems. This may involve drawing diagrams, using algebraic equations, or providing geometric proofs.

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