# Geometry

## Educators

### Problem 1

Draw a circle and several parallel chords. What do you think is true of the midpoints of all such chords?

Amrita B.

### Problem 2

Draw a circle with center $O$ and a line $\stackrel{\leftrightarrow}{T S}$ tangent to $\odot O$ at $T .$ Draw $\overline{O T}$ and use a protractor to find $m \angle O T S$ .

Georgiann A.

### Problem 3

a. Draw a right triangle inscribed in a circle.
b. What do you know about the midpoint of the hypotenuse?
c. Where is the center of the circle?
d. If the legs of the right triangle are 6 and 8, find the radius of the circle.

Amrita B.

### Problem 4

Plane $Z$ passes through the center of sphere $Q$ .
a. Explain why $Q R=Q S=Q T$ .
b. Explain why the intersection of the plane and the sphere is a circle. (The intersection of a sphere with any.plane passing through the center of the sphere is called a great circle of the sphere.)

Georgiann A.

### Problem 5

The radii of two concentric circles are 15 $\mathrm{cm}$ and 7 $\mathrm{cm} .$ A diameter $\overline{A B}$ of the larger circle intersects the smaller circle at $C$ and $D .$ Find two possible values for $A C$ .

Amrita B.

### Problem 6

For each exercise draw a circle and inscribe the polygon in the circle.
A rectangle

Georgiann A.

### Problem 7

For each exercise draw a circle and inscribe the polygon in the circle.
A trapezoid

Amrita B.

### Problem 8

For each exercise draw a circle and inscribe the polygon in the circle.
An obtuse triangle

Georgiann A.

### Problem 9

For each exercise draw a circle and inscribe the polygon in the circle.
A parallelogram

Amrita B.

### Problem 10

For each exercise draw a circle and inscribe the polygon in the circle.
An acute isosceles triangle

Georgiann A.

### Problem 11

A quadrilateral $P Q R S$ , with $\overline{P R}$ a diameter

Amrita B.

### Problem 12

For each exercise draw $\odot O$ with radius $12 .$ Then draw radii $\overline{O A}$ and $\overline{O B}$ to form an angle with the measure named. Find the length of $\overline{A B}$ .
$$m \angle A O B=90$$

Georgiann A.

### Problem 13

For each exercise draw $\odot O$ with radius $12 .$ Then draw radii $\overline{O A}$ and $\overline{O B}$ to form an angle with the measure named. Find the length of $\overline{A B}$ .
$$m \angle A O B=180$$

Amrita B.

### Problem 14

For each exercise draw $\odot O$ with radius $12 .$ Then draw radii $\overline{O A}$ and $\overline{O B}$ to form an angle with the measure named. Find the length of $\overline{A B}$ .
$$m \angle A O B=60$$

Georgiann A.

### Problem 15

For each exercise draw $\odot O$ with radius $12 .$ Then draw radii $\overline{O A}$ and $\overline{O B}$ to form an angle with the measure named. Find the length of $\overline{A B}$ .
$$m \angle A O B=120$$

Amrita B.

### Problem 16

Draw two points $A$ and $B$ and several circles that pass through $A$ and $B$ . Locate the centers of these circles. On the basis of your experiment. complete the following statement:
The centers of all circles passing through $A$ and $B$ lie on _____.
Write an argument to support your statement.

Georgiann A.

### Problem 17

$\odot Q$ and $\odot R$ are congruent circles that intersect at $C$ and $D . \overline{C D}$ is called the common chord of the circles.
a. What kind of quadrilateral is $Q D R C ?$ Why?
b. $\overline{C D}$ must be the perpendicular bisector of $\overline{Q R} .$ Why?
c. If $Q C=17$ and $Q R=30$ , find $C D$

Amrita B.

### Problem 18

Draw two congruent circles with radii 6 each passing through the center of the other. Find the length of their common chord.

Georgiann A.

### Problem 19

$\odot P$ and $\odot Q$ have radii 5 and 7 and $P Q=6 .$ Find the length of the common chord $\overline{A B}$ . (Hint: $A P B Q$ is a kite and $\overline{P Q}$ is the perpendicular bisector of $\overline{A B}$ . See Exercise $28,$ page $193 .$ Let $N$ be the intersection of $\overline{P Q}$ and $\overline{A B},$ and let $P N=x$ and $A N=y$ . Write two equations in terms of $x$ and $y . )$

Amrita B.
Draw a diagram similar to the one shown, but much larger. Carefully draw the perpendicular bisectors of $\overline{A B}$ and $\overline{B C}$ .