Curriculum / Math / 10th Grade / Unit 5: Polygons and Algebraic Relationships / Lesson 8
Math
Unit 5
10th Grade
Lesson 8 of 15
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Algebraically verify diagonal relationships in quadrilaterals and parallelograms.
The core standards covered in this lesson
G.CO.C.11 — Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
G.GPE.B.4 — Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
The foundational standards covered in this lesson
G.CO.C.9 — Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Below is parallelogram $${ABCD}$$.
$$E$$ is the midpoint of diagonal $${\overline {AC}}$$.
Prove that $$\overline{BE}\cong \overline{ED}$$.
Find the midpoints of each of the sides of the quadrilateral below to create a new quadrilateral with the midpoints as the vertices.
What do you notice about the quadrilateral formed by these midpoints? We call this the inscribed quadrilateral.
A Midpoint Miracle, accessed on June 4, 2018, 1:04 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
What particular shape has perpendicular and congruent diagonals?
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
Using the parallelogram below,
a) Write the coordinates of the point where the diagonals intersect. b) Algebraically show that the diagonals bisect each other.
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Next
Calculate and justify the area and perimeter of polygons and composite shapes off the coordinate plane.
Topic A: Distance on the Coordinate Plane
Use the Pythagorean Theorem to calculate distance on a coordinate plane. Develop a formula for calculating distances.
Standards
G.GPE.B.7
Partition horizontal and vertical line segments into equal proportions on a number line.
G.GPE.B.6
Divide a line segment on a coordinate plane proportionally and identify the point that divides the segment proportionally.
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Topic B: Classify Polygons using Slope Criteria and Proportional Line Segments
Algebraically and using the Pythagorean Theorem, determine the slope criteria for perpendicular lines.
G.GPE.B.5G.SRT.C.8
Describe and apply the slope criteria for parallel lines.
G.GPE.B.5
Identify and create parallelograms, rectangles, rhombuses, and squares on the coordinate plane.
G.GPE.B.4
Algebraically verify midsegment, median, and parallel line relationships in triangles.
G.CO.C.10G.SRT.B.4
G.CO.C.11G.GPE.B.4
Topic C: Area and Perimeter On and Off the Coordinate Plane
Calculate and justify the area and perimeter of parallelograms and triangles on the coordinate plane.
Calculate and justify composite and irregular areas on the coordinate plane.
G.GPE.B.7N.Q.A.3
Describe how the area changes when a figure is scaled.
Solve area applications by creating and solving quadratic equations.
A.CED.A.1
Describe a polygon on the coordinate plane using a system of inequalities.
A.CED.A.3G.CO.A.2
Calculate and justify the area and perimeter of polygons on the coordinate plane given a system of inequalities.
A.REI.D.12G.GPE.B.4G.GPE.B.7
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