Fascination with Pyramids again...

Do you recall the post about pyramids from last April? This is a continuation of that investigation and is a problem that has appeared on standardized tests. There are several approaches and the learning objectives for geometry students are many:


(1) Develop spatial reasoning
(2) Review terminology of space figures, pyramids in particular
(3) Make connections to the real-world problem of finding the height of an Egyptian pyramid
(4) Apply the Pythagorean Theorem or special right triangles
(5) Justify (prove) one's methods

STUDENT/READER PROBLEM
The figure attempts to depict a special regular square pyramid.
The 4 lateral edges and the 4 base edges all have the same length x.
Show that the height PT of the pyramid has length x(√2/2).

Notes:
(a) It may be instructive to encourage students to approach this by more than one method. One could ask students to find 30-60-90 as well as 45-45-90 triangles in the pyramid (lines may need to be constructed of course).
(b) Many students may assume ΔPTS is 45-45-90. Challenging them to prove it is an important objective here (there's more than one way).
(c) One could begin with a specific value of x, such as x = 10 (see the original pyramid post).
(d) I strongly urge you to have students research the Great Pyramid of Giza. Is it approximately a regular square pyramid? Are its faces equilateral triangles as in this post? There are many classic math problems associated with this Wonder of the World and it's not all about geometry!