* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Informal Geometry
Survey
Document related concepts
History of geometry wikipedia , lookup
Dessin d'enfant wikipedia , lookup
Penrose tiling wikipedia , lookup
Perspective (graphical) wikipedia , lookup
Golden ratio wikipedia , lookup
Line (geometry) wikipedia , lookup
Apollonian network wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Rational trigonometry wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Euclidean geometry wikipedia , lookup
Incircle and excircles of a triangle wikipedia , lookup
Transcript
Math 181 Spring 2000 Informal Geometry Solutions to exam I Dr. A. Bart 1. Draw and label the following geometric gures: ABC; PQR; a circle with center C and radius R, polygon MNOPQR. Everyone got this right! The answers are mainly in chapter 2. 2. Suppose that P is parallel to Q and that R is parallel to S. Use your knowledge of angles to nd the measures of angles 1; 2; 3 and 4. (Do 6 NOT measure them! You will get the wrong answer.) This is problem 4 on page 46. 1 = 31o ; 2 = 86o ; 3 = 63o ; 4 = 86o . 6 6 6 6 Figure 1: Diagram for question 2 3. a. Dene what a diagonal is. A diagonal is a segment connecting two non-consecutive (non-adjacent) vertices. b. How many diagonal does a triangle have? Explain carefully. Zero, all pairs of vertices are consecutive. In other words we can't connect nonadjacent vertices, because any two vertices must necessarily already be adjacent to one another. 1 4. Draw all the diagonals in the quadrilaterals below. a. In which quadrilaterals are the diagonals congruent? In the square, the rectangle and the isosceles trapezoid. b. In which of the quadrilaterals are the diagonals perpendicular? In the rhombus, the kite and the square. Figure 2: Question 4 5. Decide whether the following statements are true or false. Carefully explain your reasoning. a. Every right triangle is an isosceles triangle. FALSE. A right triangle has one right angle, but there is no reason to expect two of the sides to be congruent. So it is not true that every right triangle is isosceles. (Some are: 45-45-90 triangles are isosceles, but these are the only ones!) b. Every equilateral triangle is an isosceles triangle. TRUE. In an equilateral triangle all sides are congruent, so if all three are congruent, then so are two of them. So the triangle satises the requirements for isosceles triangles. c. Every square is a parallelogram. TRUE. In a square opposite sides are congruent and parallel, hence the square 2 is a parallelogram. d. Every square is a rhombus. TRUE. In a square, all sides have equal length and opposite sides are parallel. So it's a rhombus. 6. What is the relationship between rhombuses, parallelograms and kites? Explain. Every rhombus is a parallelogram, but not every parallelogram is a rhombus. So the set of rhombuses is a subset of the set of parallelograms. Every rhombus is also a kite, but not all kites are rhombuses. And kites and parallelograms have nothing else in common (only the rhombuses). 7. a. How many medians does a triangle have? Where do the medians intersect? Any triangle has 3 medians. They always intersect in the interior of the triangle: in the centroid. b. Draw an acute and an obtuse triangle and draw the medians for each. Figure 3: Medians 8. a. Dene what the altitude of a triangle is. The altitude is a line passing from a vertex , perpendicular to the opposite side (or extension of the opposite side). b. Draw an acute and an obtuse triangle and draw the altitudes. 9. a. Dene what an angle bisector is. An angle bisector is a ray which divides an angle into two congruent angles. b. Draw an angle, and use your compass to construct the angle bisector. See Appendix B page 336. 3 Figure 4: Altitudes 10. Decide if the following statements are True (T) or False (F). No explanation necessary. T Three non-colinear points uniquely determine a plane. T If three points are colinear, then they are also coplanar. T The oor and the ceiling represent parallel planes. T The wall is perpendicular to the ceiling. F Any two walls are perpendicular. 4