MAT 360: Geometric Structures
Spring 2010

## Homework assignments for MAT 360

### Assignment 10. Due date April 29

From the textbook: 356, 358, 378, 395, 425.

### Assignment 8. Due date April 8

1. Find a line intersecting two given lines and parallel to a third one. How does the number of such lines depend on the given three lines?

2. Let AB and CD be skew lines. Prove that midpoints of the segments AC, AD, BC and BD form a parallelogram, and that its plane is parallel to the lines AB and CD.

3. Find a line perpendicular to two given skew lines. How many such lines exist.

4. Find the locus of points equidistant in the space from two given points.

5. Find the locus of points in the space equidistant from given three points which do not lie on the same line.

### Assignment 7. Due date March 25

From the textbook: 305, 306, 314, 320.

### Assignment 6. Due date March 18

1. Given three lines l, m, n meeting at one point and a point A on l, construct a triangle ABC such that the lines l, m, n are its bisectors.

2. Construct a regular triangle having vertices on three given parallel lines l, m, n.

3. Construct a quadrilateral ABCD such that its diagonal AC is the bisector of its angle A and the sides are congruent to given four segments.

4. Construct a triangle with given angle A, side a opposite to A, and altitude ha dropped from A.

5. Construct a triangle with given side a, difference b-c between the other two sides and angle B opposite to the side B.

### Assignment 5. Due date March 4

From the textbook: 231, 236.

Although it is not included into the home assignement, prepare yourself to the midterm on March 4.

Take a look at the practice midterm1 and revisit the following sections of the textbook: 35, 36, 40, 42-45, 48-52, 70-73, 75-81, 83-87, 93, 95, 104-108, 111-113, 122-124, 126.

### Assignment 4. Due date February 23

Solutions for the following problems should be made of the following parts:
1. Analysis (assume that the required figure has been constructed, construct auxiliary figures, study their properties.)
2. Construction (describe the required construction as an algorithm: what constructions should be done first, second, etc.)
3. Proof (prove that the construction gives the figure with the desired properties, that is prove the properties)
4. Research (investigate, under what conditions the desired figure exists, how the number of figures with the desired properties exist for data in different ranges, etc.)

1. Given circles c and d, segment s and line l, construct a segment AB such that A lies on c, B lies on d, AB || l and AB is congruent to s.

2. Given a circle c, line l and point O, construct a segment AB such that A lies on c, B lies on l and O bisects AB.

From the textbook: 211 (a), (c), 222.

### Assignment 3. Due date February 16

180, 183, 189, 199, 203.

### Assignment 2. Due date February 9

1. Prove that in a triangle ABC, vertices A and B are equidistant from the median CM.

From the textbook: (The problems are reproduced below)
92. The sum of the medians of a triangle is smaller than its perimeter but greater than its semi-perimeter.
94. The sum of segments connecting a point inside a triangle with its vertices is smaller than the semi-perimeter of the triangle.
103. A line and a circle can have at most two common points.
(Hint: if A, B, C are distinct intersection points, then the center O of the circle is equidistant from A and B, and also from A and C).
106. Prove that two perpendiculars to the sides of an angle erected at equal distances from the vertex meet on the bisector.
100. Prove that the bisector of an angle is its axis of symmetry.
124. Construct a right triangle, given an acute angle and the hypotenuse.
149. Through a given point, construct a line making a given angle with a given line.
150. Prove that if the bisector of one of the exterior angles of a triangle is parallel to the opposite side, then the triangle is isosceles.

### Assignment 1. Due date February 2

1. Read pages 1-18 of the textbook (contained in the the sample pages). List all the encounters of implicit usage of the properties of congruences.
One of the properties is formulated explicitly on page 2, lines 2-5: One can superimpose a plane on itself or any other plane in a way that takes one given point to any other given point...

Other properties were formulated in the first lecture:
1. One can superimpose a plane on itself or any other plane in a way that takes one given ray to any other given ray.
2. A plane can be superimposed on itself keeping all the points of a given straight line fixed. This can be done in a unique way.

In other words,
1. There exists an isometry which maps a plane onto itself or any other plane in a way that takes one given ray to any other given ray.
2. There exists a unique non-identity isometry of a plane onto itself keeping all the points of a given straight line fixed.

Thus the task is to find all the places in the first 18 pages of the textbook, where these properties are used implicitly, that is without explicit mentioning. Present a solution in the form of a table with lines:
page number, line number, the property, how the property is used.

The rest of the assignemt is the following problems from the textbook: 61, 63, 67.