Download Geometry RSH Mr. Carman / Mr. Barnett Problem Set 1

Name: ______________________________________________________
Mr. Carman / Mr. Barnett
Geometry RSH
Problem Set 1
1) (NMT 2010 Grade 10 #9) The diagram shows an equilateral triangle inscribed in a circle of radius 8. The area of the
shaded region is calculated and expressed in the form
b√c. If a, b, and c are all integers and c is prime,
compute a + b + c.
2) (NMT 2010 Grade 10 #13) In the diagram, each side of square ABCD measures 12 cm. BCE is an isosceles right
triangle with base BC. Compute the number of square centimeters in the area of pentagon ABECD.
A
B
E
D
C
3) (NMT 2010 Grade 9 #14) Square ABCD is inscribed in square EFGH, which is inscribed in square JKLM. If JM = 17, JH
= 5, CH = 4, and q = the area of the shaded regions and p = the area of the unshaded regions compute p – q
F
K
L
A
B
E
4) (AMC 2004 10B #16) Three circles of radius 1 are externally tangent to each other and internally tangent to a larger
circle. What is the radius of the large circle?
5) (AMC 2004 10A #16) The 5 x 5 grid shown contains a collection of squares with sizes from 1 x 1 to 5 x 5. How many
of these squares contain the black center square?
B
6) (MC #1) Find the area of the shaded region if the square has side length 2. (Picture shows a circle and a
circumscribed square.)
Ext) Repeat using 4 circles, and then 9 circles. Remember, the side of the square is still 2.
7) (MC #2) To send the point (c, d) twice as far away from the origin, plot (2c, 2d). To send the point (c, d) three times
as far from the origin, plot (3c, 3d). Find a formula to send the point (a, b) twice as far away from ANY point.
8) (MC #3) The find the midpoint of a line segment with endpoints
,
and
,
, we plot the point
. Find a formula for plotting the “tri-point” of a line segment.
,
9) (MC #3b) The previous problem can be repeated in 3 dimensions. To find the midpoint of a line segment with
endpoints
,
,
and
,
,
, we plot the point
,
,
. Find a formula for plotting the “nth-
point) of a line segment in three dimensions.
*10) (MC #4) It is easy to construct √2 using a straightedge and compass (see diagram below). Construct √5 using a
straightedge and compass.
*11) (MC #5) The diagram on the left shows a square inscribed in a larger square by connecting the midpoints of the
larger square’s sides. The Diagram on the right is similar, but the larger square’s sides have been cut at the 1/3 mark.
Find the area of both shaded squares.
*12) (MC #6) Do you know how to read an analog clock (like the one on the wall)? At 1:00, the hour hand sits at 1,
and the minute hand sits at 12. At what time will the minute hand catch up to the hour hand? Do you think the
answer is 1:05? THINK AGAIN!!! Find the real time when the minute hand will reach the hour hand for each of the 12
hours.
13) (MC #7) Both squares below have side length 4. Find the area of both shaded regions.
*14) (MC #8) The Pythagorean Theorem states that in a right triangle,
. The Law of Cosines works for any
2
. Use the Law of Cosines to find the missing sides of the following triangles:
triangle. It states:
C
C
5
A
5
r
B
r
A
B
15) (MC #9) Find the area of the square in terms of r. Then find the area of the five little squares in terms of r.
r
*16) (MC #10) The area of an equilateral triangle with side length s is:
√
. Find the area of the shaded region.
17) (MC #11) The big white circle has a radius of r. Find the area of the shaded region in terms of r.
r
*18) (MC #12) A cylinder has a height of 8 and a diameter of 3. The label of the can has a (width-less) stripe that
wraps exactly once around the can from top to bottom. What is the length of the stripe? What about if the stripe
wrapped exactly twice around the can?
19) (MC #13) A “perfect heart” can be drawn as two congruent intersecting lines, and two congruent semi-circles.
Find the distance around the following heart.
9cm
9cm
40
20) (MC #14) The classic “five point star” contains a regular pentagon in its center. Find the perimeter of the
pentagon in the star shown below: (A to B is 5cm) ***HINT: The angle at each point of the star is 36 degrees.
A
5cm
B
C
*21) (MC #15) Find the radius of the circle inscribed in the following equilateral triangle. Each side of the triangle is 10
cm.
22) (MC #16) In the following bulls-eye, the inner circle has a radius of r. The next ring has a radius of 2r. The next ring
has a radius of 3r, and so on. Find the area of the shaded region in terms of r.
23) (2007 AMC 10B #18)
24) (2006 AMC 10A #23)
*25) (2002 AMC 10A #5)
26) (2003 AMC10A #23)
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For
example, in the figure we have 3 rows of small congruent equilateral triangles, with 5 small triangles in the base row.
How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle
consists of 23 small equilateral triangles?
*27) (MC # 17) The diagram shows an equilateral octagon within a square. Find the perimeter of the octagon if the
square has a side length of 1 inch.
*28) (GB # 1) A 4 sided convex polygon has 2 diagonals. A 5 sided convex polygon has 5 diagonals. How many
diagonals are there in a 6 sided convex polygon? What about a 13 sided convex polygon?