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Learning Math Home
Geometry Session 1: Solutions
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Solutions for Session 1, Part H

See solutions for Problems: H1 | H2 | H3 | H4

Problem H1


Pick a vertex, and then find midpoints of the two sides opposite the chosen vertex. Draw a line segment between the vertex and each of the midpoints. Repeat for the other three vertices.


Find the midpoints of the four sides. Connect the consecutive midpoints to form the four midlines.


Pick an angle (vertex). Fold the paper along the line containing the vertex and such that the two sides emanating from the vertex overlap. The crease created bisects the angle. Repeat for the other three angles (vertices).


Pick a side. Fold the paper so that the endpoints of the side overlap. The crease created defines the perpendicular bisector of the chosen side. Repeat for the other three sides.


If the quadrilateral is a rectangle, you are done, since its sides are its altitudes. If not, extend its sides in case those lines are needed. Pick a vertex and a side (or extended side) opposite it. Make a fold along a line that contains the vertex such that the two parts of the (extended) side opposite it overlap. Repeat for the other three vertices.

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Problem H2

For some quadrilaterals (specifically those which can be inscribed in a circle), the concurrency of perpendicular bisectors holds. For all quadrilaterals, the midlines come in pairs that are parallel. For some quadrilaterals (specifically those which can have a circle inscribed in them), the angle bisectors are concurrent.

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Problem H3

A shadow of a square can be a non-square. It can also be a non-rectangle. Yes, angles in a shadow of a square can be different from 90°.

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Problem H4

Shapes 1, 2, 6, 7, and 8 can cast a square shadow. One way to visualize this is to think of a non-right square pyramid. Think of a light source as being at the top vertex of the pyramid, and the edges that emanate from it as rays of light. Therefore, any cross section of the pyramid created by a plane that does not intersect the plane containing the base can cast a square shadow (the base). The shapes that might be formed by these cross sections must have four sides, because the pyramid has four sides and must not have an interior angle greater than 180°.

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