Geometry problems

1. Two houses are located on opposite sides of a straight road. The shortest distances from the houses to the road are 2 km and 3 km respectively. The distances between the two points on the main road closest to each house are 7 km. How big is the distance between the two houses?

2. By placing on each side in a regular octagon a circular arc whose center point angle is 60 degrees, a flower-like figure is formed (see the figure showing 1/8 of the figure). Determine its area if the side of the octagon is 5 cm.

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3. In a cube, the 8 corners are removed as follows: The center points on each side that go out from the corners are marked out and then the corner is sawn off in such a way that an equilateral triangle is formed with the three excellent points as corners in the triangle (see figure on the next page). The body sides now consists of six squares and 8 triangles. This body is called a cube octahedron. Calculate the volume of a cube octahedron using side a.

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4. The port cities P and N are located on opposite sides of the Atlantic. P and N have the same latitude, 40.75 degrees north. The longitude for N is 74 degrees west and for P it is 8.67 degrees west. If you are going to fly or take a boat from P to N, you may think that you should always go west, but that is not the shortest way! How much shorter in km will the journey be if you instead take the shortest route (ie along a great circle)? We assume that the earth is a sphere with a radius of 637 km. Clue: You must use the cosine theorem on a sphere.

5. One way to divide a given distance AB into three equal parts is to form a rectangle ABCD, see figure. You choose the route BC yourself. Select the center point, E, on the CD. The diagonal AC and EB intersect at a point F. The normal to CD through F intersects AB at a point G.

Show that BG = AB / 3

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