We included HMH Into Math Grade 6 Answer Key PDF Module 11 Lesson 2 Graph Polygons on the Coordinate Plane to make students experts in learning maths.
HMH Into Math Grade 6 Module 11 Lesson 2 Answer Key Graph Polygons on the Coordinate Plane
I Can graph the given vertices of a figure and determine the coordinates of an unknown vertex to complete the figure given the classification of the polygon.
Spark Your Learning
Marisol is designing a vegetable garden. The coordinate plane shows the area where the garden will be planted where 1 grid square equals 1 square meter. She has 24 meters of fencing to put around her garden. The garden can be any shape as long as she uses all of the fencing to make a border around the garden. Use the coordinate plane to show a possible design for her vegetable garden. Name the coordinates of the corners of the garden. Explain your reasoning.
Answer:
Turn and Talk What do you notice about the coordinates of the corners of the rectangles? Explain.
Answer:
Build Understanding
A vertex is the point where two sides of a polygon intersect. The plural of vertex is vertices.
Connect to Vocabulary
A polygon is a closed plane figure formed by three or more line segments that intersect only at their endpoints.
Question 1.
Look at the figures shown.
A. What do you notice about the first figure? Is it a polygon?
Answer:
It is a closed figure and it is a pentagon. And moreover, it is a polygon.
B. What do you notice about the second figure? Is it a polygon?
Answer: Not a polygon.
If starting point and the endpoint of a figure are not the same, then the figure is an open figure. A polygon is open when the segments do not all connect at the beginning and end. That is, if we draw the polygon starting at one point, we finish drawing at a different point.
C. What do you notice about the third figure? Is it a polygon?
Answer:
A circle is a closed figure but it is not a polygon. As a circle is curved, it cannot be formed from line segments, as this does not fit the conditions needed to be a polygon.
You can classify polygons by the number of sides they have. Triangles, quadrilaterals, pentagons, and hexagons are all examples of polygons.
Question 2.
Describe the polygon with vertices F(2, 5), G(7, 1), H(2, -6), and J(-3, 1).
A. Plot the points on the coordinate plane.
Answer:
To plot the point from an ordered pair on the coordinate plane:
– Start at the origin and move left/right until you get to the integer that matches the x-coordinate.
– From there, move up/down until you match up with the value on the y-axis that matches the y-coordinate.
– Mark the point and label it with the ordered pair.
B. Connect the points by drawing straight lines from F to G, G to H, H to J, and J to F. Classify the polygon you drew by the number of sides.
Answer:
– First, plot each point on the coordinate grid and then connect the lines.
– Next, in order to determine what kind of shape it is, first counts the number of vertices.
– This figure has four, so it is one of the four-sided shapes.
– Then, look to see how many of the sides are equal.
– In this case, side FG=HJ and FJ=GH. Since there are two sets of equal sides, this is either a rectangle or a parallelogram.
– Finally, check the angles.
– In this shape, the angles are right angles, so this is a rectangle.
C. How many vertices does it have?
Answer:
A two-dimensional rectangle has four vertices.
Turn and Talk Is there another way to classify the figure you drew in Part B? Explain.
Answer: yes.
– we can determine the figure by following the number of vertices given.
– First, count the number of vertices.
– In this case, there are four vertices. If there are four vertices, then the shape has four sides.
– Next, determine which shapes are possible.
– The four-sided figures are square, rhombus, trapezoid, rectangle, and parallelogram.
– Then, plot the vertices on a coordinate plane.
– Then, determine the properties of the shape.
– In this case, the shape has two pairs of sides which are equal and all its angles are right angles. Additionally, the shape has four right angles. The rectangle is the shape that meets this criterion.
– The answer is a rectangle.
Step It Out
Question 3.
An architect is drawing a plan for part of a bridge that makes a right triangle. The coordinates of two vertices of the right triangle are A(-3, 5) and 8(2, 4). Sides AC and BC will form the right angle of the triangle. What are the coordinates of Vertex C?
A. Plot Points A and B on the coordinate plane.
Answer:
To plot the point from an ordered pair on the coordinate plane:
– Start at the origin and move left/right until you get to the integer that matches the x-coordinate.
– From there, move up/down until you match up with the value on the y-axis that matches the y-coordinate.
– Mark the point and label it with the ordered pair.
The above-given vertices are A(-3, 5) and B(2, 4)
B. What must be true about a right triangle?
Answer:
– A right-angled triangle is a type of triangle that has one of its angles equal to 90 degrees. The other two angles sum up to 90 degrees. The sides that include the right angle are perpendicular and the base of the triangle. The third side is called the hypotenuse, which is the longest side of all three sides.
– The three sides of the right triangle are related to each other.
C. Is there more than one possible location for Vertex C? What are the possible coordinates of C?
Answer: No
– There is no more than a possible location for vertex C. Because a rectangle has 3 sides.
According to the properties of the right-angled triangle:
– One angle is always 90° or right angle.
– The side opposite angle of 90° is the hypotenuse.
– The hypotenuse is always on the longest side.
– The sum of the other two interior angles is equal to 90°.
– The other two sides adjacent to the right angle are called base and perpendicular.
So, the possible coordinates of C are (-3, -2)
D. Plot Point C and draw Triangle ABC.
Answer:
– First, plot the vertices and connect them.
– Next, count the number of vertices.
– There are three, so this is a three-sided figure, which you can see from its shape.
– Then, look at the sides.
– A right triangle is a three-sided closed shape, that has one perpendicular side called the leg or height of the triangle.
E. At which vertex is the right angle?
Answer:
A right angle is defined as the angle made by two rays at a vertex exactly equal to 90 degrees. That means, both the rays are perpendicular to each other.
According to the above definition:
A vertex is a right angle.
F. Draw a second right triangle in the fourth quadrant of the coordinate plane. What are the coordinates of each vertex of the triangle?
Answer:
– First, plot the vertices and connect them.
– Next, count the number of vertices.
– There are three, so this is a three-sided figure, which you can see from its shape.
– Then, look at the sides.
– A right triangle is a three-sided closed shape, that has one perpendicular side called the leg or height of the triangle.
– The coordinates in quadrant IV would be (2, -2)
G. Label the vertices of your triangle with letters. What is the vertex of the right angle in your triangle?
Answer:
The vertices of a triangle are:
A (-3, 5); B(2, 4); C(2, -2)
In the above coordinate plane, we had drawn the vertex C in the fourth quadrant.
So the right angle will be at vertex B.
H. Are the right triangles you drew polygons? Explain.
Answer:
Yes, right triangles are polygons.
A minimum of three line segments is required to connect end to end, to make a closed figure. Thus a polygon with a minimum of three sides is known as Triangle and it is also called 3-gon. An n-sided polygon is called n-gon
I. Can you plot three points on a coordinate plane that cannot be the vertices of a triangle? If so, how?
Answer:
No, we cannot plot the points.
Wherever we plot the third vertex we definitely connect the vertices and it forms a triangle.
For example, the graph can be shown below:
Turn and Talk Can you make a right triangle ABC with a right angle at Vertex B instead of at Vertex C? How could you find where to place Point C?
Answer:
The above-given vertices are A (-3, 5); B(2, 4)
Now we have to place the vertex C
Then the vertex C is (2, -2)
Question 4.
An architect is designing a plan for a new building inspired by the NyKredit Krystallen building in Copenhagen, Denmark. The building will have faces that are parallelograms without right angles. The architect plots three of the four corners of one face at Q(-5, -2), R( 1, -2) and 5(-2, 3). What are the coordinates of Point T if T is the fourth corner of that face of the building?
A. Plot Points Q, R, and S on the coordinate plane.
Answer:
To plot the point from an ordered pair on the coordinate plane:
– Start at the origin and move left/right until you get to the integer that matches the x-coordinate.
– From there, move up/down until you match up with the value on the y-axis that matches the y-coordinate.
– Mark the point and label it with the ordered pair.
The above-given vertices are Q(-5, -2), R( 1, -2) and S(-2, 3)
B. What must be true about a parallelogram?
Answer:
It is always true that a parallelogram:
– The opposite sides are parallel and congruent
– The opposite angles are congruent
– The consecutive angles are supplementary
– If any one of the angles is a right angle, then all the other angles will be at the right angle
– The two diagonals bisect each other
– Each diagonal bisects the parallelogram into two congruent triangles
– The Sum of squares of all the sides of a parallelogram is equal to the sum of squares of its diagonals. It is also called parallelogram law
C. Is there more than one possible location for Point T? What are the possible coordinates of T?
Answer:
Yes, we can plot at (3, 3) and (4, 3)
But we choose the vertex T as (4, 3)
D. Plot Point T and draw the parallelogram.
Answer:
First, plot each point on the coordinate grid and then connect the lines.
– Next, in order to determine what kind of shape it is, first counts the number of vertices.
– This figure has four, so it is one of the four-sided shapes.
– Then, look to see how many of the sides are equal.
– In this case, side ST=QR and SQ=TR. Since there are two sets of equal sides, this is either a rectangle or a parallelogram.
– Finally, check the angles.
– In this shape, the angles are right angles, so this is a parallelogram.
Check Understanding
Question 1.
You can classify polygons by the number of sides they have.
A. Graph the points A(-3, 4), B(-4, -2), C(1, -2), and D(0, 4). Connect A to B to C to D to A. What names can you use to describe this polygon? Explain.
Answer:
To plot the point from an ordered pair on the coordinate plane:
– Start at the origin and move left/right until you get to the integer that matches the x-coordinate.
– From there, move up/down until you match up with the value on the y-axis that matches the y-coordinate.
– Mark the point and label it with the ordered pair.
The above-given vertices are A(-3, 4), B(-4, -2), C(1, -2), and D(0, 4)
– If the legs or the non-parallel sides of the trapezium are of equal length, then it is called an isosceles trapezium.
-An isosceles trapezium is a trapezium in which the non-parallel sides are equal in measure. In other words, the bases are parallel and the legs are equal in measure.
– Only one pair of sides are parallel. AB ∥ CD
– Non-parallel sides (legs) are equal in measure. AD = BC
– The diagonals are equal in measure. AC = BD
– The base angles are equal in measure. ∠D = ∠C
B. Use the points you graphed in Part A. What are the possible coordinates of a Point E if ADE is a right triangle where Angle D is the right angle? Explain.
Answer:
Now take the points of ADE.
The vertices are A(-3, 4); D(0, 4)
Now we have to find out the point E so that we can form a right triangle. And moreover, angle D is the right angle. So, the point E would be (0, -2)
The coordinate plane with the right angle can be shown as:
On Your Own
Question 2.
Wesley designs a shark fin for a costume. He chooses the points K(1, 7), L(3, 5), M(6, -3), N(-5, -2), and P(-1, 1) to model the fin.
A. Graph the points on the coordinate plane.
Answer:
To plot the point from an ordered pair on the coordinate plane:
– Start at the origin and move left/right until you get to the integer that matches the x-coordinate.
– From there, move up/down until you match up with the value on the y-axis that matches the y-coordinate.
– Mark the point and label it with the ordered pair.
The above-given vertices are K(1, 7), L(3, 5), M(6, -3), N(-5, -2), and P(-1, 1)
B. Connect the points in order from K to L to M to N to P, and back to K.
Answer:
– Connect the points
– Now the graph is shown below:
C. How many sides does the polygon that you drew have? Classify the polygon by the number of sides.
Answer:
– The polygon that we drew has 5 sides.
– The five-sided polygon is called a pentagon polygon. When all the five sides of the polygon are equal in length, then it is called a regular pentagon otherwise an irregular pentagon.
– Therefore, the polygon we drew is an irregular pentagon.
Construct Arguments For Problems 3-4, tell whether the figure is a polygon. If it is a polygon, give the number of sides and classify it. If it is not, explain why not.
Question 3.
Answer:
– A two-dimensional shape which is enclosed by a finite number of straight lines joining in the form of a closed loop is called a polygon. The line segments which make the polygon are known as the polygon’s sides or edges. Whereas the corner or the point where any two sides join is called the vertex of the polygon.
– The above-given figure is a polygon.
– It has 7 sides and 7 vertices which is called a heptagon.
Question 4.
Answer:
The above-given shape is not a polygon.
– These shapes are all not polygons because they have curved sides.
– There are more irregular shapes that are also non-polygons:
– The following figures are not polygons because they are not formed by segments.
– If they are not closed figures then also they are not polygons.
– Some of them have sides that aren’t closed, some have curved sides, and one has overlapping sides.
For Problems 5-6, use the coordinate plane.
Question 5.
Consider the points K(1, 2) and L(3, 5). What possible coordinates for Point J will make Figure JKL a right triangle with the right angle at Point J? Graph Triangle JKL on the coordinate plane shown.
Answer:
– First, plot the vertices and connect them.
– Next, count the number of vertices.
– There are three, so this is a three-sided figure, which you can see from its shape.
– Then, look at the sides.
– A right triangle is a three-sided closed shape, that has one perpendicular side called the leg or height of the triangle.
– The given vertices are K(1, 2); L(3, 5). Now we have to find out the vertex J so that a right-angled triangle can be formed. The vertex J would be (3, 2).
– The coordinate plane can be shown below having J as right angle:
Question 6.
Use Tools An isosceles triangle has two sides of the same length. Isosceles triangle DEF has vertices D(4, -5) and E(-2, -5). What are possible coordinates for Point F in the section of Quadrant IV shown in the coordinate plane? Explain how you could use a ruler to check your answer.
Answer:
– First, plot the vertices and connect them.
– Next, count the number of vertices.
– There are three, so this is a three-sided figure, which you can see from its shape.
– Then, look at the sides.
– An Isosceles triangle is a triangle that has two equal sides. Also, the two angles opposite to the two equal sides are equal. In other words, we can say that “An isosceles triangle is a triangle which has two congruent sides”.
– The given vertices are D(4, -5); E(-2, -5). Now we have to place the point F in quadrant IV.
– The graph is shown below and the point F would be (1, -1):
check out the properties:
– As the two sides are equal in this triangle, the unequal side is called the base of the triangle
– The angles opposite to the two equal sides of the triangle is always equal
– The altitude of an isosceles triangle is measured from the base to the vertex (topmost) of the triangle
– A right isosceles triangle has a third angle of 90 degrees.
For Problems 7-8, use the coordinate plane.
Question 7.
A rhombus is a parallelogram with sides of equal length. The coordinates of three vertices of a rhombus are V(-2, -4), W(3, -4) and X(-6, -1). What are the coordinates of Point Y if Y is the fourth vertex of the rhombus?
A. Plot the Points V, W, and X on the coordinate plane.
Answer:
To plot the point from an ordered pair on the coordinate plane:
– Start at the origin and move left/right until you get to the integer that matches the x-coordinate.
– From there, move up/down until you match up with the value on the y-axis that matches the y-coordinate.
– Mark the point and label it with the ordered pair.
The above-given vertices are V(-2, -4); W(3, -4) and X(-6, -1)
B. What are the coordinates of Point Y? Plot Point Y on the coordinate plane.
Answer:
To plot the point from an ordered pair on the coordinate plane:
– Start at the origin and move left/right until you get to the integer that matches the x-coordinate.
– From there, move up/down until you match up with the value on the y-axis that matches the y-coordinate.
– Mark the point and label it with the ordered pair.
The above-given vertices are V(-2, -4); W(3, -4) and X(-6, -1)
The point Y would be (-1, -1) then the rhombus will form.
C. To check that all four sides are the same length, first use the coordinate plane to find the lengths of Sides VW and XY. What do the sides measure?
Answer:
To check all four sides are having the same length there is one thing we followed in the coordinate plane.
– The point V to point W we counted the squares in between them.
– The squares in between the points are 5.
– Likewise, we measured the length for point X and point Y.
– The squares in between the points X and Y are 5.
– The sides let us know the length if they are equal or not.
– The measurement of the length is 5 units
D. Next, mark off a scale of grid units on the edge of a sheet of paper and use it to find the lengths of Sides XV and YW. What do the sides measure?
Answer:
Now connect the points and find the sides of XV and YW
A rhombus is a special case of a parallelogram, and it is a four-sided quadrilateral. In a rhombus, opposite sides are parallel and the opposite angles are equal. Moreover, all the sides of a rhombus are equal in length, and the diagonals bisect each other at right angles.
– The point X to point V we counted the squares in between them.
– The squares in between the points are 4.
– Likewise, we measured the length for point Y and point W.
– The squares in between the points Y and W are 4.
– The sides let us know the length if they are equal or not.
– The measurement of the length is 4 units
Question 8.
Open-Ended Graph a square on the coordinate plane with sides of length 3 units.
Answer:
– First, count the number of vertices.
– In this case, there are four vertices. If there are four vertices, then the shape has four sides.
– Next, determine which shapes are possible.
– The four-sided figures are square, rhombus, trapezoid, rectangle, and parallelogram.
– Then, plot the vertices on a coordinate plane.
– Then, determine the properties of the shape.
– In this case, the shape has two sets of parallel lines and all the sides are the same length. Additionally, the shape has four right angles. The square is the shape that meets this criterion.
– The answer is a square.
– The length of a side is given 3 units.
– The graph is shown below:
The coordinates will be A(3, 3); B(-3, 3); C(-3, -3); D(3, -3)
Question 9.
Construct Arguments Carla drew the figure on the coordinate plane shown. Classify the figure with as many terms as possible. Explain why you can use each term.
Answer:
– Carla drew 8 sides and 8 vertices
– An octagon is a closed two-dimensional figure with eight sides, eight vertices and eight interior angles. If all the sides and interior angles of an octagon are of equal measure, then it is called a regular octagon otherwise an irregular octagon.
– It is an irregular octagon.
I’m in a Learning Mindset!
What tools can I use to solve Problem 6?
Answer:
Lesson 11.2 More Practice/Homework
Construct Arguments For Problems 1-2, tell whether the figure is a polygon. If it is a polygon, give the number of sides and classify it. If it is not, explain why not.
Question 1.
Answer:
The above-given shape is not a polygon.
– These shapes are all not polygons because they have curved sides.
– There are more irregular shapes that are also non-polygons:
– The following figures are not polygons because they are not formed by segments.
– If they are not closed figures then also they are not polygons.
– Some of them have sides that aren’t closed, some have curved sides, and one has overlapping sides.
Question 2.
Answer:
– A two-dimensional shape which is enclosed by a finite number of straight lines joining in the form of a closed loop is called a polygon. The line segments which make the polygon are known as the polygon’s sides or edges. Whereas the corner or the point where any two sides join is called the vertex of the polygon.
– The above-given figure is a polygon.
– It has 5 sides and 5 vertices which is called a pentagon.
Question 3.
Points A(3, 3), B(-6, 1), and C(-3, 1) are three vertices of a parallelogram.
A. Plot the points on the coordinate plane.
Answer:
To plot the point from an ordered pair on the coordinate plane:
– Start at the origin and move left/right until you get to the integer that matches the x-coordinate.
– From there, move up/down until you match up with the value on the y-axis that matches the y-coordinate.
– Mark the point and label it with the ordered pair.
The above-given vertices are A(3, 3), B(-6, 1), and C(-3, 1)
B. Find one possible point in the part of the coordinate plane shown that could be the fourth Vertex D of the parallelogram. Give its coordinates.
Answer:
A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure. Also, the interior angles on the same side of the transversal are supplementary. The Sum of all the interior angles equals 360 degrees.
According to the diagram, the coordinates of point D would be (0, 3)
C. Plot Point D and draw the parallelogram on the coordinate plane.
Answer:
To plot the point from an ordered pair on the coordinate plane
– The above-given vertices are A(3, 3), B(-6, 1), C(-3, 1) and D(0, 3)
– Next, in order to determine what kind of shape it is, first counts the number of vertices.
– This figure has four, so it is one of the four-sided shapes.
– Then, look to see how many of the sides are equal.
– In this case, side AB=DCand AD=BC. Since there are two sets of equal sides, this is either a rectangle or a parallelogram.
– Finally, check the angles.
– In this shape, the angles are right angles, so this is a parallelogram (given)
Question 4.
Reason Suppose you want to give the coordinates of three points that are vertices of a right triangle. How can you do this without looking at a coordinate plane?
Answer:
Question 5.
A right Triangle XYZ has Vertices X(-2, -1) and Y(1, 1) and a right angle at Vertex Z. What could the coordinates of Vertex Z be? Use the coordinate plane provided.
Answer:
– First, plot the vertices and connect them.
– Next, count the number of vertices.
– There are three, so this is a three-sided figure, which you can see from its shape.
– Then, look at the sides.
– A right triangle is a three-sided closed shape, that has one perpendicular side called the leg or height of the triangle.
– The given coordinates are X(-2, -1), Y(1, 1) and we need to find out the Z.
According to the right-angled triangle shape:
the coordinate of Z would be (-1, 1)
Now the graph is shown below:
Test Prep
Question 6.
Graph the points (1, 3), (5, -4), and (-4, 2) and connect them to form a polygon.
Answer:
To plot the point from an ordered pair on the coordinate plane:
– Start at the origin and move left/right until you get to the integer that matches the x-coordinate.
– From there, move up/down until you match up with the value on the y-axis that matches the y-coordinate.
– Mark the point and label it with the ordered pair.
The above-given vertices are (1, 3), (5, -4), and (-4, 2)
The polygon we got after connecting the vertices is an obtuse triangle.
– An obtuse-angled triangle or obtuse triangle is a type of triangle whose one of the vertex angles is bigger than 90°. An obtuse-angled triangle has one of its vertex angles as obtuse and other angles as acute angles i.e. if one of the angles measures more than 90°, then the sum of the other two angles is less than 90°.
Question 7.
Graph the points A(-3, -1), B(-3, -3), C(4, -3), and D(4, -1) on the coordinate plane. Connect the points in order from A to D. What names can you use to describe this polygon? Select all that apply.
(A) quadrilateral
(B) parallelogram
(C) square
(D) rectangle
(E) trapezoid
(F) pentagon
Answer: Option D is the correct.
To plot the point from an ordered pair on the coordinate plane:
– Start at the origin and move left/right until you get to the integer that matches the x-coordinate.
– From there, move up/down until you match up with the value on the y-axis that matches the y-coordinate.
– Mark the point and label it with the ordered pair.
The above-given vertices are A(-3, -1), B(-3, -3), C(4, -3), and D(4, -1)
– First, plot each point on the coordinate grid and then connect the lines.
– Next, in order to determine what kind of shape it is, first counts the number of vertices.
– This figure has four, so it is one of the four-sided shapes.
– Then, look to see how many of the sides are equal.
– In this case, side AD=BC and CD=AB. Since there are two sets of equal sides, this is either a rectangle or a parallelogram.
– Finally, check the angles.
– In this shape, the angles are right angles, so this is a rectangle.
Question 8.
Diana is drawing a parallelogram on a coordinate plane. She plots vertices at (-2, -2), (2, -2), and (0, 1). Select all the possible coordinates for the
fourth vertex.
(A) (4, 1)
(B) (1, 0)
(C) (-3, 1)
(D) (-4, 1)
(E) (0, -5)
(F) (2, -5)
Answer: Option A and D are correct.
A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure. Also, the interior angles on the same side of the transversal are supplementary. The Sum of all the interior angles equals 360 degrees.
– According to the definition count the squares in the coordinate plane and mark the point.
Spiral Review
Question 9.
The population of a certain bacteria doubles every 24 hours. How many times as great is the population after 5 days as the population at the start?
Answer:
The population of certain bacteria be X
In 24 hours the bacteria doubles. It means 2X.
after 5 days the growth rate will be Y.
we all know that 5 days = 120 hours
Therefore, after 5 days the growth rate will be 10X.
Question 10.
A farmer gets paid $3.75 per bushel of corn. How much does the farmer get paid for c bushels of corn? Use p to represent the farmer’s pay and write an equation that represents this situation.
Answer:
The farmer gets per bushel of corn = $3.75
P be the representation of farmer’s pay
The equation to represent the situation is:
P= $3.75 x c