This building was dedicated in 1886 (Hyde Park was annexed into Chicago in 1889). It opened to strudents in 1894. C. W. French, Principal, describes the proposed building in the 1893 Hyde Park High School Yearbook: "There will be two floors above the basement, with a large hall, forming a third story, which will be larg enough for all the public exercses of the school. A gymnasium, with a hardwood floor, will occupy a part of the basement and first story. "
Source: Alice Fluegge kindly directed me to the web site of W. H. Ray Elementary School. which now occupies this 1894 building. The web site features an illustrated article from the Hyde Park Historical Society: http://www.ray.cps.k12.il.us/history-of-ray/history-of-ray.htm
The Chicago Historical Society "American Memories" collection includes a 1912 photograph of Hyde Park High School under construction at South Stony Island Avenue (not this building). This Chicago Historical Society web page about the rivalry of Englewood High School and Hyde Park High School has an image of this building labeled as Englewood High School.