Download Geo #1 solutions

Category 2 Geometry Meet #3 – Practice #1- Solutions
1) How many diagonals does a Pentagon have? (5x2)/2 = 10/2 = 5
a hexagon ? (6x3)/2 = 18/2 = 9
a septagon ? (7x4)/2 = 28/2 = 14
an octagon? (8x5)/2 = 40/2 = 20
a nonagon? (9x6)/2 = 54/2 = 27
a decagon ? (10x7)/2 = 70/2 = 35
a dodecagon(12 sides)? (12x9)/2 = 108/2 = 54
2) What would the formula be for the number of diagonals in an n-gon?(n-sided polygon)? [n(n-3)]/2
3) A convex polygon has 170 diagonals. How many sides does it have?
[n(n-3)]/2 = 170
[n(n-3)] = 340
20x17 = 340 so n = 20 sides
4) A regular polygon has an interior angle of 160 degrees. How many sides does it have?
Interior = 160° so Exterior = 20°. If the exterior = 20°, then there are 360/20 = 18 sides.
5) A regular polygon has an interior angle between 152 and 153 degrees. How many sides does it have?
180 – 152 = 28. 360/28 ≈ 12.8 sides and 180 – 153 = 27. 360/27 ≈ 13.333 sides.
Therefore the polygon must have 13 sides.
6) The number of degrees in the exterior angle of a regular polygon is one-third that of its interior angle.
How many sides does the polygon have? Since they are supplementary, x + (1/3)x = 180 degrees
so (4/3)x = 180 and x = 135(the interior angle). The exterior is then 180-135 = 45 and 360/45 = 8 sides
7) To get to the city Joe has to drive 50 miles due East and then 120 miles due South. How many mile
50 miles
shorter would it be if Joe could drive in a straight line to the city?
Using the pythagorean theorem 502 + 1202= J2
2500+14400 = J2
120 miles
16900 = J2
130 = J
Also note this is the 5-12-13 triple all multiplied by 10
8) Triangle ABC is a right angle with angle B being the right angle. AC has a length of 17cm and AB
has a length of 15cm. How many centimeters are in the length of BD ?
A
AB2 + BC2 = AC2
The area of the triangle is
2
2
2
15 + BC = 17
(8x15)/2 = 120/2 = 60
-->
or (BD x 17)/2 = 60
225 + BC2 = 289
BC2 = 64 --> BC = 8
BD = 120/17= 7 1/17
B
F
9)
In the diagram at the right ABFE and ACDG are both squares,
and triangles BAG and EAC are congruent. All segments are whole
numbers. The combined area of the two squares is 169 m2. How
B
many meters are in the perimeter of hexagon BFECDG.
Since the sum of the areas of the two squares is 169, then both BG and EC
are equal to the square root of 169 or 13. Since all sides are whole numbers,
the triangles sides must make up a Pythagorean triple with hypotenuse equal
to 13. 5 – 12 – 13 is the only such one. So the small square has sides 5 and the
large square has sides 12. The perimeter is :
12+13+5+5+13+12 = 60
D
C
E
A
C
G
D