Chapter 4 Review Continued for Geometry Proofs Quiz

Problem 1

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
The following statements have all appeared in the activities or exercises of this or the previous chapters. You have already constructed Sketchpad diagrams illustrating these statements, and may already have written informal proofs of some of these statements. Write out a careful step-by-step proof for each statement.
a. Let $O$ be the center of a circle, and let $P, Q$ and $R$ be points on the circle. Prove that the measure of the central angle $\angle P O R$ is twice the measure of the inscribed angle $\angle P Q R$.
b. Let $O$ be the center of a circle, and let $P R$ be
a diameter of this circle. If $Q$ is a point on the circle, prove that angle $\angle P Q R$ is a right angle.
c. A median divides its triangle into two equal areas.
d. The three medians of a triangle are concurrent at a point called the centroid, often denoted as $G .$
e. The three altitudes of a triangle are concurrent at a point called the orthocenter, often denoted as $H$
f. The perpendicular bisectors of the three sides of a triangle are concurrent at a point called the circumcenter, often denoted as $O$
g. The circumcenter, $O,$ of a triangle is the center of a circle that passes through the three vertices of the triangle.
h. The opposite interior angles of a convex cyclic quadrilateral are supplementary.

Problem 2

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Create truth tables to verify that the contrapositive, $\neg Q \rightarrow \neg P,$ and the method of proof by contradiction, $(P \wedge \neg Q) \rightarrow$ false, are each equivalent to the implication $P \rightarrow Q$

Problem 3

Problem 4

Problem 5

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Draw a Venn diagram illustrating how cyclic quadrilaterals fit into the set of quadrilaterals. Your diagram should make it clear which quadrilaterals - kites, parallelograms, rectangles,rhombi, squares, and trapezoids- are always cyclic, sometimes cyclic, or never cyclic.

Problem 6

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Prove that a tangent line to a circle is perpendicular to a radius of that circle at the point of tangency. (This is the converse of the result proved by contradiction on page $93 .$ )

Problem 7

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Write a short essay explaining why it is impossible to construct a tangent to a circle from a point interior to the circle.

Problem 8

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Let $A B C D$ be a convex cyclic quadrilateral.
a. Prove that the interior angles at $A$ and $C$ are supplementary.
b. Prove that the exterior angle at $B$ is congruent to the interior angle at $D$
c. How does this situation change if $A B C D$ is not convex?

Problem 9

Problem 10

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
State the converse of the statement of Exercise $9 .$ Is this converse true or false? Explain.

Problem 11

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Prove that the perpendicular bisectors of a cyclic quadrilateral are concurrent if and only if the quadrilateral is cyclic. (Note: There are two statements to prove.)

Problem 12

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
In Activity $2,$ page $86,$ you explored the concurrency of the perpendicular bisectors of the sides of a quadrilateral. Do the same for the angle bisectors of the angles of a quadrilateral. Make conjectures, and present deductive arguments supporting (proving) your conjectures.

Problem 13

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Let WXYZ be a cyclic quadrilateral. ( $W X Y Z$ is not necessarily convex.) Let $P$ be the point on the intersection of the lines containing sides $W X$ and $Y Z$
a. If $P$ is exterior to the circle, prove that $$ P W \cdot P X=P Z \cdot P Y $$
b. Under what conditions will $P$ be interior to or on the circle?
c. Prove or disprove that $$ P W \cdot P X=P Z \cdot P Y $$

Problem 14

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Prove Ptolemy's Theorem: If $A B C D$ is a cyclic quadrilateral, then the product of the diagonals is equal to the sum of the products of the opposite sides. In symbols, $A C \cdot B D=$ $A B \cdot C D+B C \cdot D A .$ (Hint: Locate point $E$ on $A C$ so that $\angle A B E \cong \angle D B C$ and construct the segment $B E .$ Then look for similar triangles.)

Problem 15

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Given a triangle $\triangle X Y Z$, extend sides $X Y$ and $X Z$ creating exterior angles at $Y$ and $Z$.
a. Prove that the bisectors of the exterior angles at $Y$ and $Z$ are concurrent with the bisector of the interior angle at $X .$ Call this point of intersection $R$
b. Construct a circle centered at $R$, which is tangent to line $\overleftarrow{Y Z}$. $\overrightarrow{X Y}$ and $\overrightarrow{X Z} .$ This circle is an excircle of $\triangle X Y Z$.

Problem 16

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
The three excircles of $\triangle A B C$ will be tangent to the (non-extended) sides at three points. Prove that the Cevians joining the vertices of the triangle with these points of tangency are concurrent. This is the Nagel point of $\triangle A B C .$ (Compare this to Exercise 46 on page 79.)

Problem 17

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
In Sketchpad, construct two circles. Use the method described in the text to construct the radical axis of these two circles. Vary your original circles and observe how the radical axis behaves.

Problem 18

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Suppose that two circles intersect at two points $P$ and $Q .$ Prove that the radical axis of these circles is the line $\overrightarrow{P Q}$.

Problem 19

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Suppose that two circles are tangent at point $T$. Prove that the radical axis of these circles is the line through $T$ that is tangent to both circles.

Problem 20

Problem 21

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
If two circles are congruent-that is, if they have the same radius- prove that the radical axis of these circles is the perpendicular bisector of the segment between the centers of the circles.

Problem 22

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Prove that if two circles are orthogonal, then they intersect at exactly two points and their tangents are perpendicular at both of those points.

Problem 23

Problem 24

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Given triangle $\triangle A B C$.
a. Construct the center of the circumcircle of $\triangle A B C .$ Prove that your construction is correct.
b. Explain how you can use this construction to find an entire circle given just an arc of the circle.
c. What is the smallest number, $n$, of non-collinear points that determine a unique circle? Explain how you would construct a circle given just $n$ points.

Problem 25

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Suppose that $\triangle A B C$ is equilateral. Prove that the area of its circumcircle is four times the area of its incircle.

Problem 26

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
A regular hexagon is inscribed in a circle of area 2 $\pi .$ What is the area of the hexagon? (This problem has been adapted from an example given on page 30 of the Praxis Study Guide for the Mathematics Tests, ETS, $2003 .$ )

Problem 27

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Consider circle $\mathcal{C}$ with center $\mathrm{O}$ and radius $r$ Let $A_{1} A_{2}$ be a chord of $\mathcal{C}$. Let $P$ be a point
on the line $A_{1} A_{2}$ and let $d$ denote the distance from $P$ to O.
a. Assume that $P$ lies outside of $\mathcal{C}$. Let $\overleftarrow{P T}$ be tangent to $\mathcal{C}$ at $T .$ Show that $P T^{2}=d^{2}-r^{2}$
b. Still assuming that $P$ lies outside of $\mathcal{C}$, show that the product
$$P A_{1} \cdot P A_{2}=d^{2}-r^{2}$$
(Hint: First prove this if $A_{1} A_{2}$ is a diameter.
Then show that any other line $A_{1} A_{2}$ gives the same value.)
c. If $P$ lies inside of $\mathcal{C}$, it is not possible to construct the tangent from $P$ to $\mathcal{C}$. However, it still is possible to calculate $P A_{1} \cdot P A_{2}$ Assume that $P$ lies inside of $\mathcal{C}$. Show that the product $$ P A_{1} \cdot P A_{2}=d^{2}-r^{2} $$ (Hint: Use Exercise 3.)
d. Explain how the power of $P$ with respect to a fixed circle $\mathcal{C},$ Power $(P, \mathcal{C}),$ can always be calculated by $d^{2}-r^{2}$
e. What is the significance of Power $(P, \mathcal{C})$ being positive, or zero, or negative? That is, if you know that Power $(P, \mathcal{C})$ is
$$ >0, \quad \text { or }=0, \quad \text { or }<0 $$
what do you know about the point $P ?$

Problem 28

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
Let $A B$ be the diameter of a circle, and let $C$ be another point on this circle. Construct $\triangle A B C$. Let $D$ be the foot of the perpendicular from
$C$ to $A B$
a. If $A D=x$ and $B D=y,$ prove that the altitude CD of $\triangle A B C$ has length $\sqrt{x y}$
b. If $x$ and $y$ are positive numbers, then $\frac{x+y}{2}$ is their arithmetic mean, and $\sqrt{x y}$ is their geometric mean. Show that $\sqrt{x y} \leq \frac{x+y}{2}$ (This problem has been adapted from an example given on page 22 of the Praxis Study Guide for the Mathematics Tests, $\mathrm{ETS}, 2003 .)$

Problem 29

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
The Arbelos: In Figure 4.5 on page 98 , let $T$ and
$U$ be the points where lines $R A$ and $R B$ intersect the smaller arcs of the arbelos.
a. Prove that $P R$ and $T U$ are congruent line segments.
b. Prove that $P T R U$ is a parallelogram.
c. Prove that line $T U$ is an external tangent to the two smaller circular arcs of the arbelos.

Problem 30

Give clear and complete answers to the following problems and questions. Write your explanations clearly using complete sentences. Include diagrams whenever appropriate.
The Salinon: Figure 4.5 (page 98 ) gives a diagram of a salinon. Points $P, Q, R,$ and $S$ are collinear (and in that order) with $P Q \cong R S .$ Semicircles with diameters $P Q, R S,$ and $P S$ lie on the same
side of the line $P S,$ while the semicircle with diameter $Q R$ lies on the other side of $\overleftarrow{P S}$.
a. Construct a diagram of a salinon in a Sketchpad worksheet. Make your diagram robust enough that $P Q$ stays congruent to $R S$ even if you move the points around.
b. Calculate the area of the salinon.
c. The perpendicular bisector of $P S$ is the axis of symmetry of the salinon. Construct this axis of symmetry, and let $M$ and $N$ be the points where this line intersects the semicircles on diameters $P S$ and $Q R$ respectively. Construct a circle with diameter $M N$.
d. Prove that the area of the salinon is equal to the area of the circle with diameter $M N$.

Problem 31

The following problems are more challenging.
Prove that the nine-point circle for triangle $\triangle A B C$ is tangent to the incircle of $\triangle A B C$.

Problem 32

The following problems are more challenging.
Prove that the nine-point circle of triangle $\triangle A B C$ is tangent to one of the excircles of $\triangle A B C$.

Problem 33

The following problems are more challenging.
For any triangle $\triangle A B C,$ it is claimed that the diameter of the nine-point circle is half the length of the diameter of the circumcircle of $\triangle A B C$.
a. If this claim can be proved, how will the radius of the nine-point circle compare to the radius of the circumcircle? Explain.
b. What impact will this have on the relative areas of the nine-point circle and the circumcircle?
c. Prove (or disprove) the claim.

Problem 34

The following problems are more challenging.
Prove that for any triangle $\triangle A B C$, the center of the nine-point circle lies on the Euler line of the triangle, midway between the circumcenter and the orthocenter.

Problem 35

The following problems are more challenging.
We have a method for constructing a tangent to a circle $\mathcal{C}$ from a point $A,$ which may be on or exterior to $\mathcal{C}$. (See page 90.) Suppose you have two circles, $\mathcal{C}_{1}$ and $\mathcal{C}_{2} .$ Develop a strategy for constructing a line which is tangent to both $\mathcal{C}_{1}$ and $\mathcal{C}_{2} .$ (Hint: If $r_{1}$ and $r_{2}$ are the radii of $\mathcal{C}_{1}$ and $\mathcal{C}_{2},$ respectively, it will be helpful to construct a circle with radius $\left.\left|r_{1}-r_{2}\right| .\right)$

Problem 36

The following problems are more challenging.
For triangle $\triangle A B C,$ let $a=|B C|, b=|C A|,$ and $c=|A B| .$ (So $a$ is the length of the side opposite $\angle A, \text { etc. })$ For convenience, let $s$ be the semi-perimeter $\frac{a+b+c}{2}$
a. The incircle of the triangle is tangent to the lines the three sides. Label these points of tangency as $D, E,$ and $F$ on the lines $A B, B C$ and $C A$ respectively. Find expressions for the lengths of the segments $A D, D B, B E, E C$ $C F,$ and $F A$ in terms of $a, b, c, s$
b. Consider one of the excircles for $\triangle A B C$.
This circle too is tangent to the three sides, at points $X, Y,$ and $Z$ on the lines $A B, B C$ and $C A$ respectively. Find expressions for the lengths of the segments $A X, X B, B Y, Y C$ $C Z,$ and $Z A$ in terms of $a, b, c, s$

Problem 37

Are especially for future teachers.
In the Principles and Standards for School
Mathematics [NCTM 2000, 56], the National Council for Teachers of Mathematics (NCTM) recommends that
Instructional programs from prekindergarten through grade 12 should enable all students to
$\cdot$ recognize reasoning and proof as fundamental aspects of mathematics;
$\cdot$ make and investigate mathematical conjectures;
$\cdot$ develop and evaluate mathematical
arguments;
$\cdot$ select and use various types of reasoning and methods of proof.
What does this mean for your future students?
a. The NCTM has developed specific instructional recommendations for each of
four different grade bands: Pre-K-2, 3-5, $6-8,$ and $9-12 .$ Find a copy of the Principles and Standards, and study the discussion of the Reasoning and Proof Standard [NCTM $2000,56-59]$ for one of these grade bands. Choose one of the grade levels for which you are seeking certification. What are the specific recommendations with regard to reasoning and proof for your chosen grade band?
b. Find some mathematics textbooks for these
same grade levels. How are the NCTM recommendations implemented in these textbooks? Cite specific examples.
c. Write a report in which you present and critique what you learn in studying the NCTM Reasoning and Proof Standard in light of your experiences in this course. Your report should include your answers to parts
(a) and (b).

Problem 38

Are especially for future teachers.
Design several classroom activities involving reasoning and proof that would be appropriate for students in your future classroom. Write a short report explaining how the activities you design reflect both the NCTM recommendations and what you are learning about reasoning and proof in this course (in Chapters $1-4$ ).

Problem 39

Reflect on what you have learned in this chapter.
Review the main ideas of this chapter. Describe, in your own words, the concepts you have studied and what you have learned about them. What are the important ideas? How do they fit together? Which concepts were easy for you? Which were hard?

Problem 40

Reflect on what you have learned in this chapter.
a. Describe aspects of the learning environment that helped you understand the main ideas in this chapter. Which activities did you like? Dislike? Why?
b. How are you growing in understanding your own approach to learning? How do you think this will impact your approach to your professional/personal life in the future?

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