Common Core: High School - Geometry : High School: Geometry

Study concepts, example questions & explanations for Common Core: High School - Geometry

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All Common Core: High School - Geometry Resources

6 Diagnostic Tests 114 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #17 : 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 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.

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

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.

Explanation:

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 #18 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons)

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

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

5. Connect these three labeled vertices to create a triangle

6. Connect the unlabeled vertices to create a second triangle

 

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 eight vertices 

4. Connect any six vertices making six equal sides

 

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. These are the six vertices of the hexagon

5. Connect these six vertices making six 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. These are the six vertices of the hexagon

5. Connect these six vertices making six equal sides

 

Explanation:

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. These are the six vertices of the hexagon

5. Connect these six vertices making three equal sides

 

Example Question #521 : High School: Geometry

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

Explanation:

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 #20 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons)

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

 

 

Possible Answers:

\displaystyle 2r^{2}

\displaystyle 2r

\displaystyle 4r^{2}

\displaystyle 4r

Correct answer:

\displaystyle 2r^{2}

Explanation:

Since \displaystyle r is the radius, we can extend it to represent the diameter which is \displaystyle 2r.

We know that a square is made up of two right triangles.  So the diagonal must be a product of the Pythagorean Theorem; \displaystyle a^{2}+b^{2}=c^{2}.  We should let and since we are working with a square we know \displaystyle a=b.  We can simplify this formula to be

\displaystyle a^{2}+a^{2}=(2r)^{2}

\displaystyle 2a^{2}=4r^{2}

\displaystyle a^{2}= 2r^{2}

 

If we take the square root of each side we get

\displaystyle a=\sqrt{2r}

 

The area of a square is just \displaystyle a^{2}.  So the area of this square in terms of \displaystyle r is

 \displaystyle a^{2}=(\sqrt{2r})^{2}

 \displaystyle Area=2r^{2}

 

Example Question #21 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons): Ccss.Math.Content.Hsg Co.D.13

Line \displaystyle AB is perpendicular to line \displaystyle CD where both lines are the diameter of the circle.  From this information triangle \displaystyle CDA must be a(n) _______ triangle.

Screen shot 2020 08 19 at 4.35.53 pm

Possible Answers:

acute triangle

obtuse triangle

right triangle

equilateral triangle

Correct answer:

right triangle

Explanation:

Line \displaystyle AC is perpendicular to line \displaystyle AD and is 90 degrees.  We know this because line \displaystyle CD intercepts an arc of 180 degrees.  An inscribed angle is half the measure of the intercepted arc. Therefore this must be a right triangle.

Example Question #22 : Construct Inscribed Figures (Equilateral Triangles, Squares, Regular Hexagons): Ccss.Math.Content.Hsg Co.D.13

Find the side lengths of the square inscribed in the triangle.  The area of the entire triangle is 20.

Screen shot 2020 08 20 at 8.31.59 am

Possible Answers:

\displaystyle x=\frac{8}{5}

\displaystyle x=\frac{9}{5}

\displaystyle x=\frac{5}{8}

\displaystyle x=\frac{1}{2}

Correct answer:

\displaystyle x=\frac{8}{5}

Explanation:

Since we are finding the side lengths of a square, we really only need to find one side since all sides are congruent in a square.  Recall that the formula for the area of a triangle is \displaystyle A=\frac{1}{2}bh.  We can find the area of the square by finding the area of the triangle and subtracting the area of the square.  Having the area of the square will just be one of the side lengths squared, so if we take the square root of the area of the square, we will have our answer.

First, consider the small triangle in the upper corner.  This is shown below:

Screen shot 2020 08 20 at 8.32.45 am

The area of this triangle would be 

\displaystyle A_{1}=\frac{1}{2}(x)(10-x)

\displaystyle A_{1}=\frac{1}{2}10x-x^{2}

We will consider the lower two triangles combined to be the second triangle:

Screen shot 2020 08 20 at 8.33.33 am

The area of this triangle would be

\displaystyle A_{2}=\frac{1}{2}(x)(15-x)

\displaystyle A_{2}=\frac{1}{2}15x-x^{2}

And then we need to consider a third area which is the square

Screen shot 2020 08 20 at 8.34.21 am

The area of the square would be 

\displaystyle A_{3}=x^{3}

To get the total area of the triangle, we can sum all three areas.

\displaystyle A_{total}=A_{1}+A_{2}+A_{3}

We can sub in our areas and solve for to solve this problem

\displaystyle A_{total}=(\frac{1}{2}10x-x^{2})+(\frac{1}{2}15x-x^{2})+x^{2}

\displaystyle A_{total}=5x+\frac{15}{2}x-\frac{x^{2}}{2}-\frac{x^{2}}{2}+x^{2}

\displaystyle 2A_{total}=10x+15x-x^{2}-x^{2}+2x^{2} (multiply by 2 to get rid of fractions)

\displaystyle 2A_{total}=25x

\displaystyle 2(20)=25x

\displaystyle 40=25x

\displaystyle \frac{8}{5}=x

Example Question #1 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

Choose the answer that describes a congruent triangle

Possible Answers:

Triangles with corresponding sides of the same lengths but differing corresponding angles

Triangles with 2 of the 3 corresponding sides of the same lengths but differing corresponding angles

Triangles with corresponding angles of the same measure but differing corresponding sides

Triangles with corresponding sides of the same lengths and corresponding angles of the same measures

Correct answer:

Triangles with corresponding sides of the same lengths and corresponding angles of the same measures

Explanation:

To be congruent, triangles must have all corresponding angles and sides be of the same measure.  Corresponding sides of triangles are sides of the same measure in the same positions on different triangles.  This is shown in red on the two triangles below.  Corresponding angles of triangles are angles of the same measure in the same position on different triangles.  This is shown in blue on the two triangles below.

 

 

Example Question #1 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

Choose the answer that describes similar triangles

 

Possible Answers:

Triangles with corresponding sides of the same lengths and corresponding angles of the same measures

Triangles with corresponding angles of the same measure but differing corresponding sides

Triangles with corresponding sides of the same lengths but differing corresponding angles

Triangles with 2 of the 3 corresponding sides of the same lengths but differing corresponding angles

Correct answer:

Triangles with corresponding angles of the same measure but differing corresponding sides

Explanation:

Shapes, in general, are similar when they are the same shape but have different sizes.  If they were the exact same size, they would be congruent.  To have the same shape, even when the sizes differ, the shape needs to have the same angles.  Take the two triangles below for example, they have equal angles but are different sizes.  Therefore these triangles are similar.

 

Example Question #1 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

True or False: When considering right triangles, if two right triangles have a congruent hypotenuse and a congruent leg then these triangles are congruent.

Possible Answers:

False

True

Correct answer:

True

Explanation:

If two right triangles have a congruent leg and hypotenuse we can say they have two congruent sides.  Since both of these triangles are right triangles, we also know that they have a congruent angle (their 90-degree angle).  So these two triangles have two pairs of corresponding congruent sides and one pair of corresponding congruent angles.  By SAS Theorem, these right triangles are congruent.  When using this fact that two right triangles are congruent when they have a congruent hypotenuse and a congruent leg, this is called the HL Theorem.

 

Example Question #3 : Use Congruence And Similarity Criteria For Triangles To Solve Problems And To Prove Relationships In Geometric Figures

True or False: Triangles that have all three corresponding sides equal in length can still have differing corresponding angles, therefore triangles with all three corresponding sides equal in length are not guaranteed to be either similar or congruent.

Possible Answers:

False

True

Correct answer:

False

Explanation:

The Side-Side-Side (SSS) Theorem states that two triangles are congruent if the three corresponding sides of each triangle are congruent.  If you think about it, in order for all three corresponding sides to be equal they must fit together at the same angles as well making these triangles congruent.  A detailed proof based on Philoh’s approach is below for a better explanation.  Note that there are stronger proofs than this for this theorem, but this proof is best visually for this level of understanding of the theorem.

Proof: 

(We want to show that if all three sides of two triangles are equal, then these two triangles are congruent)

Consider two triangles \displaystyle $$ABC and \displaystyle $$DEF.  Assume the following:

\displaystyle $$\bar{AB}=\bar{DE}

\displaystyle $$\bar{BC}=\bar{EF}

\displaystyle $$\bar{CA}=\bar{FD}

Since \displaystyle $$\bar{BC}=\bar{EF} we are able to rotate triangle \displaystyle $$DEF to coincide \displaystyle $$\bar{BC} and \displaystyle $$\bar{EF}.  We will just call this common line segment \displaystyle $$BC for simplification.

 

Now we will draw a line from vertex \displaystyle $$A to vertex \displaystyle $$D.  This gives us two isosceles triangles.  Recall from the definition of an isosceles triangle that the two adjacent sides and angles are equivalent.  So \displaystyle $$\angle ABC = \angle ACB and \displaystyle $$\angle DBC = \angle DCB.

We can then deduce that \displaystyle $$\angle ABD = \angle ACD. We now know that at least two sides of the two given triangles are equal and the angle enclosed by these sides is congruent between the two triangles. By SAS Theorem, these two triangles are congruent.

 

 

All Common Core: High School - Geometry Resources

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