Geometry

Harold R. Jacobs

Chapter 13

The Concurrence Theorems - all with Video Answers

Educators

Section 1

Concyclic Points

Problem 1

In the figure below, the vertices of $\triangle \mathrm{KEN}$ are on circle O.
CAN'T COPY THE FIGURE
What relation do points $\mathrm{K}, \mathrm{E},$ and $\mathrm{N}$ have to each other?

Norman Atentar

Norman Atentar

Numerade Educator

Problem 2

In the figure below, the vertices of $\triangle \mathrm{KEN}$ are on circle O.
CAN'T COPY THE FIGURE
What are $\overline{\mathrm{KE}}, \overline{\mathrm{EN}},$ and $\overline{\mathrm{KN}}$ called with respect to the circle?

Norman Atentar

Numerade Educator

Problem 3

In the figure below, the vertices of $\triangle \mathrm{KEN}$ are on circle O.
CAN'T COPY THE FIGURE
What are $\sphericalangle \mathrm{K}, \sphericalangle \mathrm{E},$ and $\sphericalangle \mathrm{N}$ called with respect to the circle?

Norman Atentar

Numerade Educator

Problem 4

In the figure below, the vertices of $\triangle \mathrm{KEN}$ are on circle O.
CAN'T COPY THE FIGURE
What relation do points $\mathrm{K}, \mathrm{E},$ and $\mathrm{N}$ have to point $O ?$

Norman Atentar

Numerade Educator

Problem 5

In the figure below, the vertices of $\triangle \mathrm{KEN}$ are on circle O.
CAN'T COPY THE FIGURE
What is circle $O$ called with respect to $\triangle \mathrm{KEN} ?$

Norman Atentar

Numerade Educator

Problem 6

In the figure below, the vertices of $\triangle \mathrm{KEN}$ are on circle O.
CAN'T COPY THE FIGURE
What is point O called with respect to the triangle?

Norman Atentar

Numerade Educator

Problem 7

Use your straightedge and compass to construct an equilateral triangle each of whose sides is 5 centimeters long.
Construct the perpendicular bisectors of all three of its sides.

Ashley High

Numerade Educator

Problem 8

Use your straightedge and compass to construct an equilateral triangle each of whose sides is 5 centimeters long.
Why would you expect the perpendicular bisectors of the sides to intersect in a common point?

Jay Patel

Numerade Educator

Problem 9

Use your straightedge and compass to construct an equilateral triangle each of whose sides is 5 centimeters long.
What kind of right triangles are formed by this construction?

Ashley High

Numerade Educator

Problem 10

Trace the following triangles and circumscribe a circle about each.
CAN'T COPY THE FIGURE

William Scherer

William Scherer

Numerade Educator

Problem 11

Trace the following triangles and circumscribe a circle about each.
CAN'T COPY THE FIGURE

William Scherer

William Scherer

Numerade Educator

Problem 12

Trace the following triangles and circumscribe a circle about each.
CAN'T COPY THE FIGURE

William Scherer

William Scherer

Numerade Educator

Problem 13

The position of the circumcenter of a triangle is determined by the measures of its angles. For example, Exercise 10 shows that the circumcenter of an acute triangle lies inside it.
Where does Exercise 11 seem to indicate that the circumcenter of a right triangle is located?

Jay Patel

Numerade Educator

Problem 14

The position of the circumcenter of a triangle is determined by the measures of its angles. For example, Exercise 10 shows that the circumcenter of an acute triangle lies inside it.
Where does Exercise 12 indicate that the circumcenter of an obtuse triangle is located?

Jay Patel

Numerade Educator

Problem 15

In the figure below, $\triangle \mathrm{SAT}$ is a $30^{\circ}-60^{\circ}$ right triangle inscribed in circle $\mathrm{K} ; \mathrm{ST}=12$ Find each of the following measures.
CAN'T COPY THE FIGURE
$m \widehat{\mathrm{ST}}$.

Norman Atentar

Numerade Educator

Problem 16

In the figure below, $\triangle \mathrm{SAT}$ is a $30^{\circ}-60^{\circ}$ right triangle inscribed in circle $\mathrm{K} ; \mathrm{ST}=12$ Find each of the following measures.
CAN'T COPY THE FIGURE
AT.

Norman Atentar

Numerade Educator

Problem 17

In the figure below, $\triangle \mathrm{SAT}$ is a $30^{\circ}-60^{\circ}$ right triangle inscribed in circle $\mathrm{K} ; \mathrm{ST}=12$ Find each of the following measures.
CAN'T COPY THE FIGURE
$\mathrm{KA}$.

Norman Atentar

Numerade Educator

Problem 18

In the figure below, $\triangle \mathrm{BAR}$ is an isosceles right triangle inscribed in circle $G ; \alpha \triangle B A R=$ 50. Find each of the following measures.
CAN'T COPY THE FIGURE
$m \widehat{\mathrm{BR}}$.

Norman Atentar

Numerade Educator

Problem 19

In the figure below, $\triangle \mathrm{BAR}$ is an isosceles right triangle inscribed in circle $G ; \alpha \triangle B A R=$ 50. Find each of the following measures.
CAN'T COPY THE FIGURE
$\mathrm{BR}$.

Norman Atentar

Numerade Educator

Problem 20

In the figure below, $\triangle \mathrm{BAR}$ is an isosceles right triangle inscribed in circle $G ; \alpha \triangle B A R=$ 50. Find each of the following measures.
CAN'T COPY THE FIGURE
$\mathrm{BG}$.

Norman Atentar

Numerade Educator

Problem 21

The figures below can be used to prove that the midpoint of the hypotenuse of a right triangle is its circumcenter.
CAN'T COPY THE FIGURE
In the first figure, point $M$ is the midpoint of the hypotenuse of right $\triangle \mathrm{ABC}$.
Why is MA = MC?

Carson Merrill

Numerade Educator

Problem 22

In the second figure, point D has been chosen on $\overrightarrow{\mathrm{BM}}$ so that $\mathrm{MD}=\mathrm{BM}$. $\overline{\mathrm{DA}}$ and $\overline{D C}$ have been drawn.
Why is ABCD a parallelogram?

Ashley High

Numerade Educator

Problem 23

In the second figure, point D has been chosen on $\overrightarrow{\mathrm{BM}}$ so that $\mathrm{MD}=\mathrm{BM}$. $\overline{\mathrm{DA}}$ and $\overline{D C}$ have been drawn.
What special type of parallelogram is ABCD?

Victor Salazar

Numerade Educator

Problem 24

In the second figure, point D has been chosen on $\overrightarrow{\mathrm{BM}}$ so that $\mathrm{MD}=\mathrm{BM}$. $\overline{\mathrm{DA}}$ and $\overline{D C}$ have been drawn.
Why is $\mathrm{BD}=\mathrm{AC} ?$

Ashley High

Numerade Educator

Problem 25

Because $\mathrm{MB}=\frac{1}{2} \mathrm{BD}$ and $\mathrm{MA}=\mathrm{MC}=$
$\frac{1}{2} \mathrm{AC},$ it follows that $\mathrm{MB}=\mathrm{MA}=\mathrm{MC}$.
Why can a circle be drawn with center $\mathrm{M}$ and radius MA that goes through points $A, B,$ and $C ?$

William Scherer

William Scherer

Numerade Educator

Problem 26

In the figure below, point $O$ is the circumcenter of $\triangle \mathrm{PKR}$ and $\overline{\mathrm{OE}}$ bisects $\overline{\mathrm{KR}}$; $\mathrm{KR}=8$ and $\mathrm{OE}=3$
CAN'T COPY THE FIGURE
Why is $\overline{\mathrm{OE}} \perp \overline{\mathrm{KR}} ?$

William Scherer

William Scherer

Numerade Educator

Problem 27

In the figure below, point $O$ is the circumcenter of $\triangle \mathrm{PKR}$ and $\overline{\mathrm{OE}}$ bisects $\overline{\mathrm{KR}}$; $\mathrm{KR}=8$ and $\mathrm{OE}=3$
CAN'T COPY THE FIGURE
Find OK.

Alisia Montesano

Alisia Montesano

Numerade Educator

Problem 28

In the figure below, point $O$ is the circumcenter of $\triangle \mathrm{PKR}$ and $\overline{\mathrm{OE}}$ bisects $\overline{\mathrm{KR}}$; $\mathrm{KR}=8$ and $\mathrm{OE}=3$
CAN'T COPY THE FIGURE
Use the appropriate trigonometric ratio and either the table on page 411 or your calculator to find $\angle \mathrm{KOE}$ to the nearest degree.

Ahmad Reda

Numerade Educator

Problem 29

In the figure below, point $O$ is the circumcenter of $\triangle \mathrm{PKR}$ and $\overline{\mathrm{OE}}$ bisects $\overline{\mathrm{KR}}$; $\mathrm{KR}=8$ and $\mathrm{OE}=3$
CAN'T COPY THE FIGURE
Find \angleKOR.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 30

In the figure below, point $O$ is the circumcenter of $\triangle \mathrm{PKR}$ and $\overline{\mathrm{OE}}$ bisects $\overline{\mathrm{KR}}$; $\mathrm{KR}=8$ and $\mathrm{OE}=3$
CAN'T COPY THE FIGURE
Find $m \widehat{\mathrm{KR}}$.

Matt Gibson

Numerade Educator

Problem 31

In the figure below, point $O$ is the circumcenter of $\triangle \mathrm{PKR}$ and $\overline{\mathrm{OE}}$ bisects $\overline{\mathrm{KR}}$; $\mathrm{KR}=8$ and $\mathrm{OE}=3$
CAN'T COPY THE FIGURE
Find $\angle \mathrm{P}$.

Debasish Das

Numerade Educator

Problem 32

CAN'T COPY THE FIGURE
Given: $\triangle$ FRG with circumcircle O.
Prove: $\triangle \mathrm{ROG}$ is isosceles.

Debasish Das

Numerade Educator

Problem 33

CAN'T COPY THE FIGURE
Given: $\triangle \mathrm{GAT}$ is inscribed in circle $\mathrm{O}$.
Prove: $\angle \mathrm{O}=2<\mathrm{A}$

William Scherer

William Scherer

Numerade Educator

Problem 34

CAN'T COPY THE FIGURE
Given: $\triangle \mathrm{BID}$ with circumcircle $\mathrm{E}$ and $\mathrm{IR}=\mathrm{RB}=\mathrm{BG}=\mathrm{GD}$
Prove: $\quad \mathrm{ER}=\mathrm{EG}$

Debasish Das

Numerade Educator

Problem 35

CAN'T COPY THE FIGURE
Given: $\triangle \mathrm{WHS}$ is inscribed in semicircle $\widehat{\mathrm{WHS}}$
Prove: $\triangle \mathrm{WIS}$ is obtuse.

Hee Yun Suh

Numerade Educator