Thales and the nested triangles

In the picture

1. 1Shared point: both lines and both triangles start here
2. 2Short parallel: the third side of the small triangle
3. 3Long parallel: same direction, twice as long as the short one
4. 4Halfway crossing: the short parallel cuts the blue line at half the distance of the long one
5. 5Small triangle: same angles as the big triangle, every side halved

Examples

• A slide’s vertical support placed halfway along it is half as tall as the support at the top
• At the same moment, a stick and a tree cast shadows in the same ratio as their heights, so you can measure the tree

Quiz: Thales and the nested triangles (Age 14)

1. 1

   Two lines start from the same point, and two parallel lines cut across them. What is true about the segments this creates?

   • They are proportional: the same multiplying number links matching segments
   • They are all exactly the same length
   • Each long segment is the short one plus the same added length
2. 2

   In this figure, how do the angles of the small triangle compare with the angles of the big triangle?

   • Only one angle is shared, the one at the starting point
   • They are exactly the same angles
   • The big triangle has bigger angles because it is bigger
3. 3

   One side of the big triangle is 3 times the matching side of the small triangle. What about the other sides of the big triangle?

   • Each is also 3 times the matching small side
   • Each is multiplied by a different number
   • Each is 3 units longer than the matching small side

The answers, each with its explanation, are in the app. This lesson has left the free week, so its quiz is part of the subscription.