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What is wrong with the proof of Theorem 1.12.4?
- Nothing actually; every step is logically correct! We could use Sketchpad to explore the various possibilities for inscribing a rectangle in a square by using the ‘dragging’ feature.

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What is the largest sphere that will pass through a triangular hole whose sides are 7 in., 8 in., and 9 in. long?

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A thin triangular-shaped iron plate is accidentally dropped into a hemispherical tank, which is 10 ft. deep and full of water. It is noticed that the iron triangle is lying parallel to the surface of the water, so it is proposed to retrieve the triangle by lowering a powerful magnet into the tank at the end of a rope. What is the minimum length of rope needed if the shortest side of the triangle is 10 ft. long and the angles of the triangle are 45, 60, and 75 degrees.

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Using Sketchpad, open a new sketch and draw a triangle D ABC . · On side BC construct the outward pointing square having BC as one of its sides; construct the center of this square and label it X. · Construct the corresponding outward pointing squares on CA and AB ; label their respective centers Y and Z.
Construct the segments AX and YZ . What properties do AX and YZ have? Check your conjecture by dragging the vertices of DABC around.

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Quadratic equations: The Greeks used geometry where now we would use algebra. For instance, they knew how to construct the roots of the quadratic equation x 2 -ax +b = 0 for given values of a,b when a 2 > 4b . Let’s use Sketchpad to illustrate their method. Draw a pair of perpendicular lines, which we’ll think of as the x- and y-axes. On the yaxis choose a fixed point and label it 1; the distance of this point from the point of intersection of the two perpendicular lines is to be thought of as specifying what unit length means. Given a, b draw the point C having (a, b) as coordinates as well as the point on the y-axis having ycoordinate 1. Now draw the circle having the line segment from this point on the y -axis to C as a diameter. Finally, label the points of intersection of this circle with the x-axis by A and B. You should have a figure looking like
Show that the x-coordinates of A, B are the roots of the equation x 2 -ax +b = 0 . Where was the condition a 2 > 4b used? What is the equation of the circle you drew? This all looks pretty straightforward to us now that we have the analytic geometry of circles available to us, but it should be remembered that almost 2,000 years elapsed after the Elements were written before Descartes combined algebra with geometry!

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A gardener cut a piece of sod to fill a hole in the shape of an acute triangle in a grass lawn. When he came to put the grass sod in the hole he found that it fit perfectly, but only with the wrong side up. To fit the sod in the triangular hole with the right side up he had to 50 cut it. How did he cut it into three pieces so that the shape of each piece was unchanged when he turned it over?
- He cut it in a form of right triangle so that it could be fit in a sod again.

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Birthday Cake: For her birthday party, Sally’s father baked a chocolate cake in the shape of a triangular prism. Sally will have eight of her friends at her birthday party, and everyone likes chocolate cake and icing. How is Sally to cut the cake efficiently so that she and each of her friends get equal shares of cake and icing?

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Using Sketchpad, in a new sketch draw any convex quadrilateral ABCD. Recall that a convex quadrilateral is one that has all interior angles less than 180° . · On side AB construct the outward pointing square having AB as one of its sides. Construct the center of this square and label it Z. · Construct corresponding squares on the other sides BC , CD , and DA , and label their centers X, U and V respectively. · Draw the line segments ZU and XV . Make a conjecture about the properties of ZU and XV . Check these properties by dragging the vertices of the quadrilateral ABCD. Drag one of the vertices so that the quadrilateral becomes concave. Do the properties of ZU and XV still hold true or do they change for concave quadrilaterals?

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Consider a piece wise linear figure consisting of a polygon containing h holes (non-overlapping polygons in the interior of the outer polygon) has a total of n edges, where n includes both the interior and the exterior edges. Express the sum of the interior angles as a function of n and h. Prove your result is true.