1.1Write the Notation for a Segment
What is the notation for the line segment with endpoints $A$ and $B$?
Solution
The notation for the segment is
$$
\overline{AB}
$$
A segment has two endpoints and finite length.
Geometry Practice
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Difficulty
Direct Practice
1.2Find a Complementary Angle
One angle measures $37^\circ$. If it is complementary to another angle, what is the measure of the other angle?
Solution
Complementary angles sum to $90^\circ$.
So the other angle is
$$
90^\circ - 37^\circ = 53^\circ
$$
1.3Use Vertical Angles
Two lines intersect. One angle measures $128^\circ$. What is the measure of its vertical angle?
Solution
Vertical angles are congruent, so the vertical angle has the same measure.
$$
128^\circ
$$
1.4Complete a Triangle Angle Sum
In a triangle, two angles measure $48^\circ$ and $67^\circ$. What is the measure of the third angle?
Solution
The angles of a triangle sum to $180^\circ$.
So the third angle is
$$
180^\circ - 48^\circ - 67^\circ = 65^\circ
$$
1.5Find the Base Angles of an Isosceles Triangle
An isosceles triangle has a vertex angle of $46^\circ$. What is the measure of each base angle?
Solution
The two base angles are congruent.
First find their total:
$$
180^\circ - 46^\circ = 134^\circ
$$
Then divide by $2$:
$$
\frac{134^\circ}{2} = 67^\circ
$$
Each base angle measures $67^\circ$.
1.6Check the Triangle Inequality
Can side lengths $5$, $7$, and $13$ form a triangle?
Solution
No. For a triangle to exist, the sum of any two side lengths must be greater than the third.
Here,
$$
5+7=12
$$
and $12$ is not greater than $13$.
So these side lengths do not form a triangle.
1.7Identify a Triangle Congruence Criterion
Two triangles have two corresponding sides and the included angle congruent. Which triangle congruence criterion applies?
Solution
That is the SAS criterion:
Side
Angle
Side
The angle must be included between the two sides.
1.8Scale a Similar Triangle
Two similar triangles have a scale factor of $4$ from the smaller triangle to the larger triangle. If a side of the smaller triangle is $6$ cm, what is the corresponding side of the larger triangle?
Solution
Multiply the smaller side by the scale factor:
$$
6 \times 4 = 24
$$
The corresponding side is $24$ cm.
1.9Use the Pythagorean Theorem
A right triangle has legs of lengths $8$ and $15$. What is the length of the hypotenuse?
Solution
Use the Pythagorean theorem:
$$
a^2+b^2=c^2
$$
So,
$$
8^2+15^2=c^2
$$
$$
64+225=c^2
$$
$$
289=c^2
$$
$$
c=17
$$
1.10Find an Inscribed Angle
An inscribed angle intercepts an arc that measures $124^\circ$. What is the measure of the inscribed angle?
Solution
An inscribed angle measures half its intercepted arc.
So,
$$
\frac{124^\circ}{2}=62^\circ
$$
The inscribed angle measures $62^\circ$.
Difficulty
Integrated Practice
2.1Solve with Parallel Lines
Two parallel lines are cut by a transversal. One corresponding angle measures $3x+7^\circ$ and the matching angle measures $5x-9^\circ$. Find $x$.
Solution
Corresponding angles are congruent, so set the expressions equal:
$$
3x+7=5x-9
$$
Subtract $3x$:
$$
7=2x-9
$$
Add $9$:
$$
16=2x
$$
Divide by $2$:
$$
x=8
$$
2.2Exterior Angle in an Isosceles Triangle
An isosceles triangle has a vertex angle of $34^\circ$. What is the measure of an exterior angle at one base?
Solution
The two base angles are congruent.
First find one base angle:
$$
180^\circ - 34^\circ = 146^\circ
$$
$$
\frac{146^\circ}{2}=73^\circ
$$
An exterior angle at the base is supplementary to the base angle:
$$
180^\circ - 73^\circ = 107^\circ
$$
The exterior angle measures $107^\circ$.
2.3Use Similarity with Perimeter
Two similar triangles have a scale factor of $3:2$ from the smaller triangle to the larger triangle. If the smaller triangle has perimeter $20$ cm, what is the larger perimeter?
Solution
Perimeters scale by the same factor as corresponding side lengths.
So multiply by $\frac{3}{2}$:
$$
20 \cdot \frac{3}{2} = 30
$$
The larger perimeter is $30$ cm.
2.4Apply a Dilation and a Reflection
A point is at $(2,-3)$. It is dilated about the origin by a factor of $4$ and then reflected across the $x$-axis. Where does it land?
Solution
First apply the dilation by factor $4$:
$$
(2,-3)\mapsto(8,-12)
$$
Then reflect across the $x$-axis, which changes the sign of the $y$-coordinate:
$$
(8,-12)\mapsto(8,12)
$$
The image is $(8,12)$.
2.5Use Parallelogram Angle Properties
A parallelogram has one angle that measures $68^\circ$. What are the measures of an adjacent angle and the opposite angle?
Solution
In a parallelogram, opposite angles are congruent and adjacent angles are supplementary.
The opposite angle is
$$
68^\circ
$$
The adjacent angle is
$$
180^\circ - 68^\circ = 112^\circ
$$
So the angles are $112^\circ$ and $68^\circ$.
2.6Find the Number of Sides of a Polygon
A polygon has an interior angle sum of $1260^\circ$. How many sides does it have?
Solution
Use the interior angle sum formula:
$$
(n-2)180^\circ = 1260^\circ
$$
Divide by $180$:
$$
n-2=7
$$
Add $2$:
$$
n=9
$$
The polygon has $9$ sides.
2.7Find the Radius from Coordinates
A circle has center $(2,-1)$ and passes through $(6,2)$. What is its radius?
Solution
The radius is the distance from the center to a point on the circle:
$$
r=\sqrt{(6-2)^2+(2-(-1))^2}
$$
$$
r=\sqrt{4^2+3^2}
$$
$$
r=\sqrt{16+9}=\sqrt{25}=5
$$
The radius is $5$.
2.8Use a Radius and a Tangent
A radius is drawn to a point of tangency, and a segment from that same point of tangency goes to an external point. If the angle at the external point is $27^\circ$, what is the angle at the center?
Solution
The radius is perpendicular to the tangent, so one angle in the triangle is $90^\circ$.
The angles of a triangle sum to $180^\circ$, so the angle at the center is
$$
180^\circ - 90^\circ - 27^\circ = 63^\circ
$$
The angle at the center is $63^\circ$.
Difficulty
Applied Problems
3.1Use Similar Triangles in a Shadow Problem
A $6$-foot person casts an $8$-foot shadow. At the same time, a tree casts a $20$-foot shadow. How tall is the tree?
Solution
The triangles are similar, so corresponding sides are proportional:
$$
\frac{6}{8}=\frac{h}{20}
$$
Solve for $h$:
$$
8h=120
$$
$$
h=15
$$
The tree is $15$ feet tall.
3.2Solve a Ladder Problem
A $13$-foot ladder reaches a window $12$ feet above the ground. How far is the base of the ladder from the wall?
Solution
Use the Pythagorean theorem with the ladder as the hypotenuse:
$$
x^2+12^2=13^2
$$
$$
x^2+144=169
$$
$$
x^2=25
$$
$$
x=5
$$
The base of the ladder is $5$ feet from the wall.
3.3Find the Area of a Sector
Find the area of a sector with radius $10$ m and central angle $72^\circ$.
Solution
Use the sector area formula:
$$
A=\frac{\theta}{360^\circ}\cdot \pi r^2
$$
Substitute the values:
$$
A=\frac{72}{360}\cdot \pi(10^2)
$$
$$
A=\frac{1}{5}\cdot 100\pi
$$
$$
A=20\pi
$$
The area is $20\pi$ square meters.
3.4Find the Surface Area of a Rectangular Prism
A rectangular prism has length $8$ cm, width $5$ cm, and height $3$ cm. What is its surface area?
Solution
Use the surface area of a rectangular prism:
$$
SA=2(lw+lh+wh)
$$
Substitute:
$$
SA=2(8\cdot 5+8\cdot 3+5\cdot 3)
$$
$$
SA=2(40+24+15)
$$
$$
SA=2(79)=158
$$
The surface area is $158$ cm$^2$.
3.5Show a Rectangle with Coordinates
Use coordinates to show that the quadrilateral with vertices $A(0,0)$, $B(4,0)$, $C(4,3)$, and $D(0,3)$ is a rectangle.
Solution
Check the side directions:
$AB$ is horizontal, so its slope is $0$.
$BC$ is vertical, so its slope is undefined.
$CD$ is horizontal, so its slope is $0$.
$DA$ is vertical, so its slope is undefined.
Horizontal and vertical lines are perpendicular, so each corner angle is $90^\circ$.
Also, opposite sides are parallel:
$AB \parallel CD$
$BC \parallel DA$
Since the quadrilateral has four right angles, it is a rectangle.
Difficulty
Challenge / Synthesis
4.1Use the Power of a Point
From an external point, a tangent segment has length $12$ cm and a secant has external part $9$ cm and whole length $x$ cm. Find $x$.
Solution
Use the tangent-secant relationship:
$$
(\text{tangent})^2=(\text{external})(\text{whole})
$$
So,
$$
12^2=9x
$$
$$
144=9x
$$
$$
x=16
$$
The whole secant length is $16$ cm.
4.2Classify a Square by Coordinates
Determine whether the quadrilateral with vertices $A(0,0)$, $B(4,2)$, $C(2,6)$, and $D(-2,4)$ is a square.
Solution
First check the side lengths:
$$
AB=\sqrt{(4-0)^2+(2-0)^2}=\sqrt{20}
$$
$$
BC=\sqrt{(2-4)^2+(6-2)^2}=\sqrt{20}
$$
Similarly, $CD=\sqrt{20}$ and $DA=\sqrt{20}$, so all four sides are equal.
Now check the slopes of adjacent sides:
$$
m_{AB}=\frac{2-0}{4-0}=\frac{1}{2}
$$
$$
m_{BC}=\frac{6-2}{2-4}=-2
$$
The slopes are negative reciprocals, so $AB \perp BC$.
Since the figure has four equal sides and a right angle, it is a square.
4.3Scale Area and Volume Under a Dilation
A figure is dilated by a factor of $3$. If its original area is $14\text{ cm}^2$ and its original volume is $14\text{ cm}^3$, what are the new area and new volume?
Solution
Under a dilation by factor $k$:
area scales by $k^2$
volume scales by $k^3$
Here, $k=3$.
New area:
$$
14\cdot 3^2 = 14\cdot 9 = 126
$$
New volume:
$$
14\cdot 3^3 = 14\cdot 27 = 378
$$
The new area is $126\text{ cm}^2$ and the new volume is $378\text{ cm}^3$.
4.4Find a Chord Length from the Center
A circle has center $O$ and radius $10$ cm. A chord $AB$ is $16$ cm long, and the segment from $O$ to the chord meets the chord at its midpoint $M$. Find the length of $OM$.
Solution
Because the segment from the center meets the chord at its midpoint, $AM=MB=8$ cm.
Then $\triangle OMA$ is a right triangle with hypotenuse $OA=10$ and one leg $AM=8$.
Use the Pythagorean theorem:
$$
OM^2+8^2=10^2
$$
$$
OM^2+64=100
$$
$$
OM^2=36
$$
$$
OM=6
$$
So $OM=6$ cm.