B = the maximum number of unique scalene triangles having all sides of integral lengths and perimeter less than 13

Transcription

1 GEOMETRY TEAM #1 A = the m C in parallelogram ABCD with m B= (4x+ 15), m D= (6x+ ) B = the degree measure of the smallest angle in triangle ABC with m A= ( x+ 0), m B= ( x+ 7), m C= (x 15) Find the value of A B. GEOMETRY TEAM # A = the number of diagonals for a regular polygon with 4 sides B = the measure, in degrees, of one interior angle in a regular dodecagon C = the sum of the exterior angles of a regular octagon, one at each vertex D = the length of the apothem in a regular hexagon having a side of 6 Find the value of A+ B C D. GEOMETRY TEAM # A = the degree measure of the smaller angle formed by the hands of a clock at 4:6 B = the maximum number of unique scalene triangles having all sides of integral lengths and perimeter less than 1 C = the sum of the possible integral lengths of the third side of a triangle having the other two sides of and 8 Find the value of A ( C+ B). GEOMETRY TEAM #4 A = the complement of the smaller angle when the same side interior angles of two parallel lines have measures of x and ( x 0) B = the length of EF when E is the midpoint of DF and DE = x, DF = 7x 7 suur suur C = the m RST in the diagram when RX TY, m R = 118, m T = 159 D = the m BAC in ΔABC when m B= 6, the exterior angle at Ais x and the exterior angle at C is 18 Find the value of ( A+ D) ( B+ C).

2 GEOMETRY TEAM #5 In the diagram, m R = 0, RS = 5, m TSP = 45 and PT is an altitude of Let the perimeter of Find the value of ( ) Δ RSP. ΔRPT be A+ B. A B. GEOMETRY TEAM #6 The degree measure of the larger angles of a rhombus are double the degree measures of the smaller angles. The shorter diagonal has a length of 10. A = the length of a side of the rhombus B = the length of the longer diagonal C = the degree measure of the larger interior angle D = the perimeter of the rhombus Find the value of C B +. D A GEOMETRY TEAM #7 A = the length of the hypotenuse of a right triangle when the sum of the squares of the lengths of the three sides of the triangle is 00 B = the perimeter of a rectangle when the width is 1 and the diagonal is 6 more than the length Find the value of A+ B. GEOMETRY TEAM #8 Consider the following statements. If the statement is true, assign it a value of. If the statement is false, assign it a value of 1. Find the sum of the values. I. Through any two points there is exactly one plane. II. Two intersecting lines are coplanar. III. At least one line can be drawn through any three points. IV. If points P and Q lie in a plane, then suur PQ lies in that plane. V. Two triangles are congruent if the measure of the three angles of one are equal to the measure of the corresponding parts of the other.

3 GEOMETRY TEAM #9 Given regular octagon ABCDEFGH. A = the length of HC if the side of the octagon is 1 B = the measure of one interior angle of the octagon C = the complement of one exterior angle of the octagon D = the area of trapezoid ABCH Find the value of A+ B C D. GEOMETRY TEAM #10 Given Δ FHJ, point K is on FJ and G is a point on HF so that GK HJ. FG = x, HJ = y, GH = 9, KJ = x +, FK = 4, GK = 5. Find the perimeter of ΔHJF expressed as a simplified improper fraction. GEOMETRY TEAM #11 A = the length of an angle bisector in an equilateral triangle with a side having a length of 8 B = the least value of diagonal AC in rectangle ABCD with a perimeter of 0 Find the value of A + B. GEOMETRY TEAM #1 A side of one equilateral triangle is congruent to the altitude of a second equilateral triangle. Find the ratio of the perimeter of the larger triangle to the perimeter of the smaller triangle. GEOMETRY TEAM #1 In the diagram, m AEC = x 6 x, m BED = x Find the m AEB expressed as a simplified mixed number.

4 GEOMETRY TEAM #14 Which of the following has the largest perimeter? Your answer will be the name of the figure, either square, or triangle, or hexagon, spelled correctly of course. 1. A square with a diagonal of 4. An isosceles triangle with legs having length and base angles of 0. A regular hexagon with apothem

5 GEOMETRY TEAM SOLUTIONS #1 90 #8 1 # 15 #9 90 # 5 # #4 58 #11 7 #5 5 #1 #6 #1 1 4 #7 5 #14 Hexagon A: 19 B: 49 A B= = 90 1 A: B and D are opposite angles so they are congruent. 4x+ 15 = 6x+, x=, m B= m D= 41. Angle C is the supplement of B and D so its measure is 19. B: The sum of the angles is x+ 1 = 180, x = 4, m A= 6, m B= 40, m C = 69. The smallest angle is A: 5 B: 150 C: 60 D: A: ( n ) A B C D + = ( ) = 15 n 4 = 1 = 5 B: One exterior angle of a 1 sided polygon is 60 = 0. One interior angle would be the 1 supplement of 0, which is 150. C: The sum of the exterior angles of any polygon, one at each vertex, is 60. D: Draw a triangle from a vertex of the hexagon to the center of the hexagon and the foot of the apothem on the side of the triangle. This is a triangle. The hypotenuse is 6, the apothem is opposite a 60 angle and is 1 times the hypotenuse,.. 5 A: 78 B: C: 40 A C B ( ) ( + ) = = 5. A: 60 h 11 m = 11 = 78 B: Triangles with side lengths of :,5,4;,4,5;,,4. C: The third side must solve 5 < x < 11which would be 6,7,8,9,10 and the sum is ( A+ D) ( B+ C) ( ) = ( ) = 58 A: These two angles are supplementary so 4x = 00, x = 50 and the smaller angle is 50. The complement is 40. B: Double DE and set it equal to DF. 4x 6 = 7x 7, x = 7, DE = EF = 11 C: Draw a line through S parallel to suur RX and TY suur. Then use same side interior angles. The angle on the upper side of the parallel is 6 and the angle on the under side is 1. The sum is 8.

6 D: Exterior + interior angle of a triangle = 180. m BCA = 4, and the sum of the remote interior angles equals the exterior angle so = xx, = 68, m BAC = = Let PT = ST = x, SP = x, RP = x. Using Pythagorean Theorem in Δ RPT, we have ( ) ( ) x x x =, and solving this gives = + = ( A B) ( ) = = x =. The perimeter is x+ x+ 5+ xwhich is 4x + 5., which simplifies to A: 10 B: 10 C: 10 D: 40 C B + = = + 0 = D A A: 10 B: 4 A+ B= 5 A: a + b + c = 00, a + b = c. Substituting c for a + b in the first equation gives c = 00, c=± 10, reject the 10. B: Let the length of the rectangle be x and the diagonal x + 6. Using Pythag, ( ) + = + =, reject 8. Length is 9, width is 1, perimeter is 4. 1 x x 6, x 9, I. is false (there is exactly one line) II is true III. is false (At least one line can be drawn through any points) IV is true V is false (these triangles are similar) After drawing the octagon and HC, draw the altitudes from A and B to HC. This gives us a trapezoid with upper base and legs all equal to 1. Since the triangles formed by drawing the altitudes are , the length of the altitude is A: The length of HC is = 1+. B: One interior angle of the octagon is 15. The exterior angle is 60 = 45 and the interior angle is the 8 supplement of that. C: The complement of an exterior angle, which is 45, is =. D: The area is ( 1 1) The value of + 1 A+ B C D= = ΔFKG ~ ΔFJH by AA therefore sides are proportional. FG FK x 4 =, =, solving gives x = 6, 7,reject the 7. This makes GH KJ 9 x FG = 4, FH = 1, KJ = 9, FJ = 1.

7 KG To find JH, =, =, y =. The perimeter of JH 1 y Δ HJF = =. 4 4 A + B A: 4 B: 5 = = 7 A: The angle bisector in an equilateral triangle is also a median and an altitude. Draw the segment and find it is opposite a 60 degree angle which would be 4. B: The perimeter is 0 which makes the sum of a width and a length 10. By trial and error, using different lengths for the width and length, such as 1 and 9, the diagonal would be 8 using and 8, the diagonal would be 68 so you can see the diagonal is getting smaller. So, trying 5 and 5 makes the diagonal Let the length of a side of one triangle be x and the altitude of the other x. In the nd triangle, the altitude is opposite a 60 degree angle so to find the side of the triangle divide by and multiply by making the side x. This is the larger side and we want the ratio of the larger to the smaller. Since the ratio of the sides is the same as the ratio of the perimeters, we have x x = These two angles are vertical so they are congruent. = ( )( + ) = x = 4or 1 x 1x 84 0, x 1 x 4 0, = + 4, 4 = 0, x x x x x 14. Hexagon Square The diagonal is 4 making the side 4 and the perimeter is 16. Triangle The legs are, drawing the altitude and it is opposite a 0 degree angle, the altitude is 1 the hypotenuse which is. Half of the base, since it is opposite a 60 degree angle, is. And the base is 6 making the perimeter Hexagon Apothem is is opposite the 60 angle, divide by to get. Double this to get the side of the hexagon that has a perimeter of 1. Which is largest? hexagon