Connor Blount
Mr. Rice
Term 4 Geometry Project
Chapters 10 and 11
Section 1
Chapter 11
11-1 Areas of Parallelograms:
Key Concept:


If a parallelogram has an area of A square units, a base of b units, and a height of h units, then
area equals the product of the base and the height.
A = bh
h
b
11-2 Areas of Triangles, Trapezoids, and Rhombi:

You have learned how to find the areas of squares, rectangles and parallelograms. The area of a
triangle is related to these formulas.
h
B
Key Concept: Area of a Triangle


If a triangle has an area of A square units, a base of b units, and a corresponding height of h
units, then the area equals one half the product of the base and the height.
A = ½ bh
Key Concept: Area of a Trapezoid


If a trapezoid has an area of A square units, bases of b1 units and b2 units, and a height of h
units, then the area equals the product of one half the height and the sum of the lengths of each
base.
A = ½ h(b1 + b2)
b1
h
b2
Key Concept: Area of a Rhombus



A
If a rhombus has an area of A square units and diagonals of d1 and d2 units, then area equals
one half the product of the length of each diagonal.
A = ½ d1d2
Example: A = ½ (AC)(BD)
B
D
C
11-3 Areas of Regular Polygons and Circles:
Key Concept: Area of a Regular Polygon


If a regular polygon has an area of A square units, a perimeter of P units, and an apothem of a
units, then the area is one-half the product of the perimeter and the apothem.
A = ½ Pa
K
J
L
P
N
Q
M
Key Concept: Area of a Circle

A = 𝜋𝑟 2
r
11-4 Areas of Composite Figures:

A composite figure is a figure that can be separated into regions that are basic figures. To find
the area of a composite figure, separate the figure into basic figures of which we can find the
area. The sum of the areas of the basic figures is the area of the composite figure.
A
C
D
E
F
B
11-5 Geometric Probability and Areas of Sectors:
Geometric Probability:

Finding the probability that a point lies in a part of a two-dimensional figure by comparing the
area of the part to the area of the whole figure.
Key Concept:

If a point in region A is chosen at random, then the probability P(B) that the point is in region B,
which is the interior of region A, is
P(B) = area of region B/area of region A
A
B
Sectors and Segments of Circles:
Key Concept: Area of a Sector

If a sector of a circle has an area of A square units, a central angle measuring N°, and a radius of
r units, then A = N/360 𝜋𝑟 2
N°
Segments:


The region of a circle bounded by an arc and a chord.
To find the area of a segment, subtract the area of the triangle formed by the radii and the
chord from the area of the sector containing the segment.
10-1 Circles and Circumferences:

A circle is the locus of all points in a plane equidistant from a given point called the center of the
circle. A circle is usually named by its center point.
A
B
D
F
E
Key Concept: Circumference

For a circumference of C units and a diameter of d units or a radius of r units, C = 𝜋𝑑 or C = 2𝜋𝑟
10-2 Measuring Angles and Arcs:

A central angle has the center of the circle as its vertex, and its sides contain two radii of the
circle
Key Concept: Sum of Central Angles


The sum of the measures of the central angles of a circle with no interior points in common is
360.
m<1 + m<2 + m<3 = 360
1
2
3
Key Concept: Arcs of a Circle
Theorem 10.1:

In the same or in congruent circles, two arcs are congruent if and only if their corresponding
central angles are congruent.
Postulate 10.1: Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
10.3 Arcs and Chords:

The endpoints of a chord are also the endpoints of an arc.
Theorem 10.2:



In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding
chords are congruent.
The chords of adjacent arcs can form a polygon. An inscribed polygon is a polygon where all the
vertices lie on the circle. Circumscribed means the polygons vertices lie on the circle.
Examples: if line AB ≅ line 𝐶𝐷, then arc 𝐴𝐵 ≅ arc 𝐶𝐷
A
B
C
D
Theorem 10.3:

In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its
arc.
Theorem 10.4:

In a circle or in congruent circles, two chords are congruent if and only if they are equidistant
from the center.
10.4 Inscribed Angles:
Theorem 10.5: Inscribed Angle Theorem


If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of
its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed
angle).
Example: m < ABC = ½ (measure of arc ADC)
A
B
D
C
Theorem 10.6:


If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the same arc,
then the angles are congruent.
Example: <DAC ≅ <DBC
A
B
D
C
Theorem 10.7:


If the inscribed angle of a triangle intercepts a semicircle, the angle is a right angle.
Example: arc ADC is a semicircle, so m < ABC =
A
D
B
C
Theorem 10.8:


If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
Example: <A and <C are supplementary.
B
C
A
D
10-5 Tangents:
Theorem 10.9:


If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of
tangency.
Example: If line RT is a tangent, then OR is perpendicular to RT
R
T
Theorem 10.10:


If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line
is tangent to the circle.
Example: If line segment OR is perpendicular to line RT, line RT is a tangent.
R
T
Theorem 10.11:


If two segments from the same exterior point are tangent to a circle, then they are congruent.
Example: 𝐴𝐵 ≅ 𝐴𝐶
B
A
C
10-6 Secants, Tangents, and Angle Measures:

A line that intersects a circle in exactly two points is called a secant.
Theorem 10.12:

If two secants intersect in the interior of a circle, then the measure of an angle formed is onehalf the sum of the measure of the arcs intercepted by the angle and its vertical angle.
Theorem 10.13:

If a secant and a tangent intersect at the point of tangency, then the measure of each angle
formed is one-half the measure of the intercepted arc.
10-7 Special Segments in a Circle
Theorem 10.15:


If two chords intersect in a circle, then the products of the measures of the segments of the
chords are equal.
Example: AE x EC = BE x ED
A
B
D
C
Theorem 10.16:


If two secant segments are drawn to a circle from an exterior point, then the product of the
measure of one secant segment and its external secant segment is equal to the product of the
measures of the other secant segment and its external secant segment.
Example: AB x AC = AE x AD
C
B
A
E
D
Theorem 10.17:

If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the
square of the measure of the tangent segment is equal to the product of the measures of the
secant segment and its external secant segment.

Example: WX x WX = WZ x WY
X
W
Z
Y
10-8 Equations of Circles
Key Concept: Standard Equation of a Circle

An equation for a circle with center at (h, k) and a radius of r units is
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
Review/Practice Questions:
1. If the length of line segment VU is 18 and arc 𝑉𝑈 ≅ 𝑎𝑟𝑐 𝑊𝑇, what is the length of line
segment WT?
W
V
T
U
2. If arc VW is 57°, what is the measure of arc YX?
V
Y
W
X
3. If AB is parallel to CD, and measure of arc AB = 60°, and the measure of arc AC is = 80°, find
the measure of arc CD.
A
B
C
B
D
4. Find the area of the triangle.
12in
14in
5. If the height is 8in and the length of the base is 9 inches, what is the area?
6. If the height is 5 inches and the top base is 4 inches long and the bottom base is 9 inches
long, what is the area?
7. If the radius is equal to 6.75in , what is the area of the circle?
8. If the height is 7.65 inches and the length of the base is 9.63 inches, what is the area?
9. If the diameter is 19 inches, what is the area of the circle?
10. If the height is 5.32 inches and the length of the base is 9.75 inches, what is the area?
Answer Key:
1. 18 : arcs are congruent
2. 57° : arcs are congruent
3. 140° : 60° + 80°
4. 84 𝑖𝑛2 : ½ (12x14)
5. 72 𝑖𝑛2 : 8x9
6. 32.5 𝑖𝑛2 : ½ (5)(4+13)
7. 143.14 𝑖𝑛2 : 6.752 𝜋
8. 73.67 𝑖𝑛2 : 7.65 x 9.63
9. 283.53 𝑖𝑛2 : (19/2)2 x 𝜋
10. 51.87 𝑖𝑛2 : 5.32 x 9.75