Senior Math Circles: Geometry I

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1 Universit of Waterloo Facult of Mathematics entre for Education in Mathematics and omputing pening Problem (a) If 30 7 = + + z Senior Math ircles: Geometr I, where, and z are positive integers, then what is the value of + + z? (b) etermine the value of (c) etermine the value of (e) etermine the value of (f) If a > 0, determine the value of a + (g) etermine all real numbers a > 0 for which a + a +. a + a + a + = a + a + a + a (d) HLLENGE Evaluate Epress our answer in the form a+b c, where a, b, c, d are integers. (Source: 995 Putnam d ompetition)

2 Universit of Waterloo Facult of Mathematics entre for Education in Mathematics and omputing Facts bout ngles Senior Math ircles: Geometr I F. ngles along a straight line add to 80. The are called supplementar angles. F2. ngles in a right angle add to 90. The are called complementar angles. F3. pposite angles are equal. Wh? ngles around a point add to 360. Wh? F4. onsider two parallel lines. orresponding angles are equal ( = ). lternating angles are equal ( = z). Interior angles are supplementar (w + = 80 ). z w F5. ngles in a triangle add to 80. an ou prove this? z

3 F6. The eterior angle of a triangle equals the sum of the other two interior angles. That is, w = + z. z w F7. What is the sum of the angles in a quadrilateral? How could we figure this out? F8. What is the sum of the angles in a pentagon? In a heagon? In a polgon with n sides? regular polgon has all angles equal and all side lengths equal. What is each angle in a regular pentagon? What is each angle in a regular heagon? What is each angle in a regular decagon? F9. The sum of the eterior angles in a pentagon is 360. an ou find two different was to show this? The sum of the eterior angles in an conve polgon is 360. e a b d c F0. triangle is called isosceles if two of its side lengths are equal and is called equilateral if all three side lengths are equal. If a triangle has two side lengths equal, then the opposite two angles are equal.

4 Facts bout Side Lengths SF. In a right-angled triangle, the Pthagorean Theorem tells us that a 2 + b 2 = c 2. an ou prove this? Pthagorean Triple is a triple (a, b, c) of positive integers with a 2 + b 2 = c 2. What Pthagorean Triples do ou know? SF2. If a triangle has two angles equal, then the two opposite sides are equal. a b c SF3. If <, then a < b. If a < b, then <. a b c SF4. If a, b and c are the side lengths of a triangle, the Triangle Inequalit tells us that b + c > a and a + c > b and a + b > c. an ou eplain wh this is true? a b c SF5. There are two kinds of special triangles. The first has angles 45, 45 and 90. The second has angles 30, 60 and 90. If the shortest side in each has length, what are the other side lengths? These can be scaled b an factor

5 Facts bout ircles F. circle is determined b its centre and its radius. circle is the set of points at a fied distance from a fied point. r F2. chord is a line segment that begins and ends on a circle. If a radius cuts a chord into two equal parts, then the radius is perpendicular to the chord. If a radius is perpendicular to a chord, then it cuts the chord into two equal parts. F3. If is a diameter, then = 90. F4. If is a chord and is on the major arc, then = 2. F5. If is a chord and and are on the major arc, then =.

6 Universit of Waterloo Facult of Mathematics entre for Education in Mathematics and omputing Problem Set Senior Math ircles: Geometr I. In the diagram, is parallel to. etermine the values of and. E 3 50 F G 3 2. Triangle has a right angle at. is etended to so that =. The bisector of angle meets at E. Prove that E = In the diagram, is parallel to and = =. If = 52, determine the measure of. 4. The diagram shows three squares of the same size. What is the value of? 5. The diagram shows a rhombus F GHI and an isosceles triangle F GJ in which GF = GJ. ngle F JI equals. What is the measure of angle JF I? F G J I H 6. is a square. The point E is outside the square so that E is an equilateral triangle. Find angle E.

7 7. The diagram shows two isosceles triangles in which the four angles marked are equal. The two angles marked are also equal. Find an equation relating and. 8. In the diagram, QSR is a straight line. QP S = 2 and P Q = P S = RS. What is the size of QP R? P Q S R 9. The diagram shows a regular nonagon with two sides etended to meet at point. What is the size of the acute angle at? 0. The three angle bisectors of triangle LMN meet at a point as shown. ngle LNM is 68. What is the size of angle LM? L N M. In the figure shown, = F and, F, F E, and E are all straight lines. etermine an equation relating, and z. F z E 2. The angles of a nonagon are nine consecutive numbers. What are these numbers? 3. What is the measure of the angle formed b the hands of a clock at 9:0? 4. etermine the sum of the angles,,,, and E in the five-pointed star shown. E 5. In P QR, P Q = P R. P Q is etended to S so that QS = QR. Prove that P RS = 3( QSR).

8 6. In the diagram, let M be the point of intersection of the three altitudes of triangle. If = M, then what is in degrees? M 7. In the diagram, P W is parallel to Q, S and T lie on Q, and U and V are the points of intersection of P W with SR and T R, respectivel. If SUV = 20 and V T = 2, what is the measure of URV? P U R V W Q S T 8. Three regular polgons meet at a point and do not overlap. ne has 3 sides and one has 42 sides. How man sides does the third polgon have? an ou find other sets of three polgons that have this propert? 9. etermine the number of different values of a for which the given triangle is isosceles. 3a 6 5a 6 3a alculate the area of figure In the diagram, which side is the longest:,,,, or?

9 22. If a 3 m stake casts a shadow 7 m long, what is the height of a tree that casts a shadow 63 m long? 23. scalene triangle is a triangle whose side lengths are all different. etermine the side lengths of all possible scalene triangles with integer side lengths and perimeter less than In the diagram, = 4, = 6, and is parallel to. If = 45, determine the length of. 25. Triangle is similar to Y Z. If = 4, Y Z = 9, and = Y = p, determine the value of p. 26. triangle can be formed having side lengths 4, 5 and 8. It is impossible however, to construct a triangle with side lengths 4, 5 and 0. Using the side lengths 2, 3, 5, 7 and, how man different triangles with eactl two equal sides can be formed? 27. triangle can be formed having side lengths 4, 5 and 8. It is impossible however, to construct a triangle with side lengths 4, 5 and 0. Ron has eight sticks, each having an integer length. He observes that he cannot form a triangle using an three of these sticks as side lengths. What is the shortest possible length of the longest of the eight sticks? 28. In the adjacent squares shown, the vertices, and lie in a straight line. What is the value of? In the diagram, = = 5, E = 8 and E = 4. etermine the length of. E 30. etermine the length of the chord if = 5 and N = 3. N

10 3. If = 0 and = 3, determine the length of N. N 32. circle has a diameter of length 26. If a chord of the same circle has a length of 0, how far is the chord from the centre? 33. alculate the distance between the parallel chords P Q and Y if P Q = 6, Y = 8, and the radius of the circle is 5. P Q Y 34. The two parallel chords and are a distance of 4 part. If has length 2 and the radius of the circle is 0, calculate the length of. 35. Two circles with centre and have radii 5 and 8, respectivel. The circles intersect at the points and Y. If Y = 8, determine the length of, the distance 5 8 between the centres. Y 36. In the diagram, P = 3 and Q = 20, where P and Q are the centres of the circles. etermine the length of if P Q = 2. P T Q 37. In the diagram, is the centre of the circle. etermine the measure of QS. P 00 Q 40 T S R

11 38. etermine the measure of etermine the measure of and of E 40. cclic quadrilateral is a quadrilateral that has all four its vertices on the same circle. Prove that opposite angles are supplementar. 4. In the diagram, is inscribed in the semicircle with centre. If =, determine the measure of. 42. In the diagram, Y Z is right-angled at Z. W is the midpoint of Y, and the circle with diameter ZW intersects W at V. If Y = 50 and W V = 7, determine the length of Z. Y W V Z 43. has its vertices on a circle, as shown. The bisector of the angle at meets the circumference at. From, a line is drawn perpendicular to the chord so that it meets the circumference at E. Prove that E is a diameter of the circle. E

12 44. In the diagram, points, P, Q, and lie on line segment. The semi-circle with diameter has centre P and the semi-circle with diameter has centre Q. The two semi-circles intersect at R. If P RQ = 40, determine the measure of R. P Q R 45. If is the centre of the circle and = 82, what is the value of in degrees? 82