Question 24
The mathematician Thales visits Egypt. Known as a brilliant Greek mathematician, he is invited to the royal court by the Egyptian king, who asks him to measure the height of the pyramid of Kheops, his country's pride.
Kheops is a pyramid with a square base. The diagonals of the square intersect at a point; the segment from that point to the apex is perpendicular to the base and is the pyramid's height.
Thales agrees. The next day he brings a ruler and begins measuring the height, astonishing the people of Egypt.
First he measures a base side as 227 m. He chooses its midpoint and walks straight outwards from the pyramid until the shadow of the top of his head coincides with the shadow of the pyramid's apex, as shown.
Thales is 1.7 m tall, and his shadow measures 2.55 m. He then measures 102 m from the midpoint of the base side to where he stands. Finally he announces the pyramid's height, to the admiration of the people.
To see how Thales determined the height, model the pyramid and his position as in the following diagrams.
H is the intersection of the base diagonals; M is the midpoint of the chosen base side. AB is Thales's height from his feet to his head, and O is the endpoint of both the apex's shadow and the shadow of his head.
1. Find HM,MA,BA and AO.
2. Are triangles BAO and SHO similar? Why? Hence find SH.
Kheops is a pyramid with a square base. The diagonals of the square intersect at a point; the segment from that point to the apex is perpendicular to the base and is the pyramid's height.
Thales agrees. The next day he brings a ruler and begins measuring the height, astonishing the people of Egypt.
First he measures a base side as 227 m. He chooses its midpoint and walks straight outwards from the pyramid until the shadow of the top of his head coincides with the shadow of the pyramid's apex, as shown.
Thales is 1.7 m tall, and his shadow measures 2.55 m. He then measures 102 m from the midpoint of the base side to where he stands. Finally he announces the pyramid's height, to the admiration of the people.
To see how Thales determined the height, model the pyramid and his position as in the following diagrams.
H is the intersection of the base diagonals; M is the midpoint of the chosen base side. AB is Thales's height from his feet to his head, and O is the endpoint of both the apex's shadow and the shadow of his head.
1. Find HM,MA,BA and AO.
2. Are triangles BAO and SHO similar? Why? Hence find SH.