### All Common Core: High School - Geometry Resources

## Example Questions

### Example Question #11 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons)

Which of the following images is a diamond that is inscribed in a circle?

**Possible Answers:**

**Correct answer:**

To determine which image illustrates a diamond that is inscribed in a circle, first understand what the term "inscribed" means.

"Inscribed" means to draw inside of. Therefore, a diamond inscribed in a circle means the diamond will be drawn inside of a circle.

The image that illustrates this is as followed.

### Example Question #161 : Congruence

Which of the following images is a circle that is inscribed in a diamond?

**Possible Answers:**

**Correct answer:**

o determine which image illustrates a circle that is inscribed in a diamond, first understand what the term "inscribed" means.

"Inscribed" means to draw inside of. Therefore, a circle inscribed in a diamond means the circle will be drawn inside of a diamond.

The image that illustrates this is as followed.

### Example Question #11 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons)

True or False: An inscribed figure is a shape that fits inside another geometric shape.

**Possible Answers:**

True

False

**Correct answer:**

True

This is true. An inscribed figure is one that fits inside another shape; it can touch the sides of the shape it is inside but it cannot cross over these sides. Below is a circle inscribed in a square. We could also say that the square is circumscribed about the circle.

### Example Question #14 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons)

How does one find the incenter of a triangle when trying to inscribe a circle within the triangle?

**Possible Answers:**

Find the angle bisectors of any two angles of the triangle. Their intersection is the incenter.

Pick any point within the triangle. Call this the incenter of the triangle.

Find the height of the triangle and draw the auxiliary line down to the base. The midpoint of this line is the incenter.

Find the bisectors of coming from any point of any two sides of the triangle. Their intersection is the incenter.

**Correct answer:**

Find the angle bisectors of any two angles of the triangle. Their intersection is the incenter.

The incenter is the intersection of the triangle’s three angle bisectors. Drawing two of these angle bisectors is sufficient enough to find the incenter. Below is a figure that illustrates the incenter of a triangle as point A.

### Example Question #13 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons): Ccss.Math.Content.Hsg Co.D.13

What are the steps to inscribing an equilateral triangle in a circle?

**Possible Answers:**

1. Make a point at any point on the circle’s circumference

2. Draw an arc across the circle with a compass that is set to the length of the radius, this will be the next vertex

3. Draw arcs in this fashion until there are six vertices

4. Label every other vertex so that there are three vertices

5. Connect these three vertices making three equal sides

1. Make a point at any point on the circle’s circumference

2. Draw a triangle with its three points on the circle’s circumference

1. Make a point at any point on the circle’s circumference

2. Draw an arc across the circle with a compass that is set to double the length of the radius, this will be the next vertex

3. Draw arcs in this fashion until there are three vertices

4. Label every other vertex so that there are three vertices

5. Connect these three vertices making three equal sides

**Correct answer:**

1. Make a point at any point on the circle’s circumference

2. Draw an arc across the circle with a compass that is set to the length of the radius, this will be the next vertex

3. Draw arcs in this fashion until there are six vertices

4. Label every other vertex so that there are three vertices

5. Connect these three vertices making three equal sides

Explanation: The steps are shown below along with figures for more explanation

1. Make a point at any point on the circle’s circumference

2. Draw an arc across the circle with a compass that is set to the length of the radius, this will be the next vertex

3. Draw arcs in this fashion until there are six vertices

4. Label every other vertex so that there are three vertices

5. Connect these three vertices making three equal sides

### Example Question #15 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons)

What are the steps to inscribing a square inside a circle?

**Possible Answers:**

1. Given a circle (or draw a circle) draw a line across the diameter

2. Draw a line perpendicular to the diameter that also bisects the diameter

3. Label all points that intersect the circumference of the circle

4. Connect the points on the outer edge of the circle to form the four sides of the square

1. Given a circle (or draw a circle) draw a line across the diameter

2. Draw a line parallel to the diameter

3. Label all points that intersect the circumference of the circle

4. Connect the points on the outer edge of the circle to form the four sides of the square

1. Given a circle (or draw a circle) draw a line across the diameter

2. Draw a line perpendicular to the diameter that also bisects the diameter

3. Create two triangles from these intersections

4. Combine the triangles to form a square

**Correct answer:**

1. Given a circle (or draw a circle) draw a line across the diameter

2. Draw a line perpendicular to the diameter that also bisects the diameter

3. Label all points that intersect the circumference of the circle

4. Connect the points on the outer edge of the circle to form the four sides of the square

Explanation: The steps are shown below along with figures for more explanation

1. Given a circle (or draw a circle) draw a line across the diameter

2. Draw a line perpendicular to the diameter that also bisects the diameter

3. Label all points that intersect the circumference of the circle

4. Connect the points on the outer edge of the circle to form the four sides of the square

### Example Question #16 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons)

Which of the following is the correct way to inscribe a square in a right triangle?

**Possible Answers:**

Bisect the right angle with a line that intersects with the hypotenuse, call this point A. Draw lines perpendicular to each leg of the right triangle that passes through point A, these are two sides of your square. The final two sides of the square are lines that run from the points intersecting the two legs down to the right angle vertex.

Bisect either leg with a line that intersects with the hypotenuse, call this point A. Draw angle bisectors through each of the acute angles.

Bisect either of the acute angles with a line that intersects with the hypotenuse, call this point A. Draw lines perpendicular to each leg of the right triangle that passes through point A, these are two sides of your square. The final two sides of the square are lines that run from the points intersecting the two legs down to the right angle vertex.

Find the center of the triangle. Draw a square around the center.

**Correct answer:**

Bisect the right angle with a line that intersects with the hypotenuse, call this point A. Draw lines perpendicular to each leg of the right triangle that passes through point A, these are two sides of your square. The final two sides of the square are lines that run from the points intersecting the two legs down to the right angle vertex.

This becomes clear when demonstrated through a picture. Not only does this allow us to inscribe a square within a right triangle, but it also gives us the largest square possible within the right triangle.

### Example Question #17 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons): Ccss.Math.Content.Hsg Co.D.13

What are the steps to inscribe a hexagon in a circle?

**Possible Answers:**

1. Make a point at any point on the circle’s circumference

3. Draw arcs in this fashion until there are eight vertices

4. Connect any six vertices making six equal sides

1. Make a point at any point on the circle’s circumference

3. Draw arcs in this fashion until there are six vertices

4. These are the six vertices of the hexagon

5. Connect these six vertices making six equal sides

1. Make a point at any point on the circle’s circumference

3. Draw arcs in this fashion until there are six vertices

4. Label every other vertex

5. Connect these three labeled vertices to create a triangle

6. Connect the unlabeled vertices to create a second triangle

**Correct answer:**

1. Make a point at any point on the circle’s circumference

3. Draw arcs in this fashion until there are six vertices

4. These are the six vertices of the hexagon

5. Connect these six vertices making six equal sides

Explanation: The steps are shown below along with figures for more explanation

1. Make a point at any point on the circle’s circumference

3. Draw arcs in this fashion until there are six vertices

4. These are the six vertices of the hexagon

5. Connect these six vertices making three equal sides

### Example Question #12 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons)

True or False: The inscribed angle theorem states that the inscribed angle is half the measure of the intercepted arc.

**Possible Answers:**

False

True

**Correct answer:**

False

This is demonstrated by the inscribed angle on a circle. An inscribed angle is an angle whose vertex is on the circumference of the circle and whose sides extend as chords of the circle. This theorem is proven to be true and can be used to solve for angles of inscribed figures in circles. This theorem is portrayed in the figure below.

### Example Question #202 : Congruence

Find the area of the inscribed square in terms of where is the radius of the circle.

**Possible Answers:**

**Correct answer:**

Since is the radius, we can extend it to represent the diameter which is .

We know that a square is made up of two right triangles. So the diagonal must be a product of the Pythagorean Theorem; . We should let and since we are working with a square we know . We can simplify this formula to be

If we take the square root of each side we get

The area of a square is just . So the area of this square in terms of is