Illustrative Math Alignment: Grade 8 Unit 2
Dilations, Similarity, and Introducing Slope
Lesson 9: Side Length Quotients in Similar Triangles
Use the following Media4Math resources with this Illustrative Math lesson.
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Math Clip Art--Geometry Concepts--Tessellation Pattern 2 | Math Clip Art--Tessellation Pattern 2TopicGeometry Concepts DescriptionThis image presents another tessellation pattern using basic shapes from pattern blocks. The basic pattern block shapes include equilateral triangles, squares, rhombuses, trapezoids, hexagons, and parallelograms. They are arranged to cover a plane without gaps or overlaps, demonstrating the principles of tessellation. Tessellations with pattern blocks help students explore geometric concepts such as symmetry, transformations, and tiling. They provide a visual and interactive way to understand how shapes can be combined to create complex patterns. |
Geometric Constructions with Polygons |
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Math Clip Art--Geometry Concepts--Tessellation Pattern 3 | Math Clip Art--Tessellation Pattern 3TopicGeometry Concepts DescriptionThis image displays a tessellation pattern created with basic shapes from pattern blocks. The basic pattern block set includes equilateral triangles, squares, rhombuses, trapezoids, hexagons, and parallelograms. These shapes are arranged to tile a plane without gaps or overlaps. Using pattern blocks for tessellations allows students to explore geometric concepts such as symmetry, transformations, and spatial reasoning. They provide a hands-on approach to understanding how different shapes can be combined to form intricate patterns. |
Geometric Constructions with Polygons |
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Math Clip Art--Geometry Concepts--Tessellation Pattern 4 | Math Clip Art--Tessellation Pattern 4TopicGeometry Concepts DescriptionThis image presents a tessellation pattern using basic shapes from pattern blocks. A basic pattern block set includes equilateral triangles, squares, rhombuses, trapezoids, hexagons, and parallelograms. These shapes are arranged to cover a plane without gaps or overlaps, illustrating the concept of tessellation. Tessellations with pattern blocks help students explore geometric concepts like symmetry, transformations, and tiling. They provide a visual and interactive way to understand how shapes can be combined to create complex patterns. |
Geometric Constructions with Polygons |
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Math Clip Art--Triangles--Two Triangles Share Vertical Angles Congruent--Labeled | Math Clip Art--Triangles--Two Triangles Share Vertical Angles Congruent--Labeled
This is from a collection of triangular shapes. They come labeled and unlabeled. |
Applications of Triangles |
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Math Clip Art--Triangles--Two Triangles Share Vertical Angles Congruent--Unlabeled | Math Clip Art--Triangles--Two Triangles Share Vertical Angles Congruent--Unlabeled
This is from a collection of triangular shapes. They come labeled and unlabeled. |
Applications of Triangles |
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Math Clip Art--Triangles--Two Triangles Share Vertical Angles Non Congruent--Labeled | Math Clip Art--Triangles--Two Triangles Share Vertical Angles Non Congruent--Labeled
This is from a collection of triangular shapes. They come labeled and unlabeled. |
Applications of Triangles |
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Math Clip Art--Triangles--Two Triangles Share Vertical Angles Non Congruent--Unlabeled | Math Clip Art--Triangles--Two Triangles Share Vertical Angles Non Congruent--Unlabeled
This is from a collection of triangular shapes. They come labeled and unlabeled. |
Applications of Triangles |
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Math Clip Art--Triangles--Two Triangles Share Vertical Angles Similar--Labeled | Math Clip Art--Triangles--Two Triangles Share Vertical Angles Similar--Labeled
This is from a collection of triangular shapes. They come labeled and unlabeled. |
Applications of Triangles |
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Math Clip Art--Triangles--Two Triangles Share Vertical Angles Similar--Unlabeled | Math Clip Art--Triangles--Two Triangles Share Vertical Angles Similar--Unlabeled
This is from a collection of triangular shapes. They come labeled and unlabeled. |
Applications of Triangles |
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Math Clip Art--Triangles--Two Triangles with Common Base--Congruent--Labeled | Math Clip Art--Triangles--Two Triangles with Common Base--Congruent--Labeled
This is from a collection of triangular shapes. They come labeled and unlabeled. |
Applications of Triangles |
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Math Clip Art--Triangles--Two Triangles with Common Base--Congruent--Unlabeled | Math Clip Art--Triangles--Two Triangles with Common Base--Congruent--Unlabeled
This is from a collection of triangular shapes. They come labeled and unlabeled. |
Applications of Triangles |
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Math Clip Art--Triangles--Two Triangles with Common Base--Non Congruent--Labeled | Math Clip Art--Triangles--Two Triangles with Common Base--Non Congruent--Labeled
This is from a collection of triangular shapes. They come labeled and unlabeled. |
Applications of Triangles |
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Math Clip Art--Triangles--Two Triangles with Common Base--Non Congruent--Unlabeled | Math Clip Art--Triangles--Two Triangles with Common Base--Non Congruent--Unlabeled
This is from a collection of triangular shapes. They come labeled and unlabeled. |
Applications of Triangles |
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Math Clip Art: Comparing Right Triangles | Math Clip Art: Comparing Right Triangles Use these clip art images and the background grid to test if the right triangles are similar. |
Right Triangles |
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Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 1 | Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 1TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are congruent. Triangle ABC has sides 5, 4, and 6, while triangle DEF also has sides 5, 4, and 6. The SSS Postulate ensures that these triangles are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 2 | Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 2TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are congruent. Triangle ABC has sides 5, 4, and 6, while triangle DEF has sides 5, 7, and 6. Because corresponding sides are not all congruent, then the triangles are not congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 3 | Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 3TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are congruent. Triangle ABC has sides 5 and 6, and triangle DEF has sides 5 and 6, but no information on the third side. As a result, there isn't enough information to know if they are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 4 | Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 4TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are congruent. Both triangles have two sides (5 and 6) and an included angle of 60°. As a results of the SAS Postulate, the triangles are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 5 | Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 5TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are congruent. Triangle ABC has sides 5 and 6 with an included angle of 60°, and triangle DEF has sides 5 and 6 with an included angle of 62°. Because corresponding angles are not congruent, then the triangles are not congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 6 | Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 6TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are congruent. Both triangles have angles of 63°, 60°, and 57°, but no information on side lengths. As a result, we can't conclude if the triangles are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 7 | Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 7TopicGeometric Shapes DescriptionDetermine if two triangles on a grid are congruent. Both triangles appear identical in shape and orientation, positioned differently on the grid. Using the grid, you can see that corresponding sides are congruent. Therefore the triangles are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 8 | Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 8TopicGeometric Shapes DescriptionDetermine if two shapes on a grid are congruent. The shapes appear as two congruent triangles within a diamond shape, with each triangle reflected across the center axis. Using the grid, you can see that corresonding sides are congruent. Therefore, the triangles are congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 9 | Math Example--Geometric Shapes--Analyzing Congruent Shapes--Example 9TopicGeometric Shapes DescriptionDetermine if two triangles on a grid are congruent. The triangles have differing shapes and orientations on the grid, suggesting different side lengths or angles. Therefore, they are not congruent. Congruence of shapes is fundamental in geometry as it allows us to establish relationships between figures and understand properties of transformations. This example collection illustrates congruence by analyzing side lengths and angles to determine equivalence. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 1 | Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 1TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10, 8, and 12, while triangle DEF has sides 5, 4, and 6. Triangles are similar if corresponding sides are proportional. Here, the ratio of corresponding sides is 2:1 (10:5, 8:4, 12:6). Thus, the triangles are similar. Therefore, the answer is Yes, the triangles are similar. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 10 | Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 10TopicGeometric Shapes DescriptionDetermine if two right triangles are similar. The triangle on the left has angle 40°, and the other triangle has a 60° angle. For similarity, all corresponding angles must be congruent. The angles in each triangle do not match, so the triangles aren't similar. Therefore, the answer is No, the triangles aren't similar. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 2 | Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 2TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10, 8, and 12, while triangle DEF has sides 5, 3, and 6. Triangles are similar if all corresponding sides are proportional. Only two pairs of sides here have a 2:1 ratio, so the triangles are not similar. Therefore, the answer is No, the triangles arenÕt similar. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 3 | Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 3TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are similar. Both triangles have angles 60°, 57°, and 63°. Triangles are similar if corresponding angles are equal. Since all corresponding angles match, the triangles are similar. Therefore, the answer is Yes, the triangles are similar. Understanding the concept of geometric shapes is essential for developing geometric reasoning. These examples demonstrate how to analyze and verify similarity between shapes by comparing corresponding sides and angles. The worked-out examples provided help students visualize and understand this key concept. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 4 | Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 4TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are similar. Triangle ABC has angles 75°, 42°, and an unknown angle, and triangle DEF has angles 75° and 63° and an unknown angles. Triangles are similar if all corresponding angles are congruent. Solving for the unknown angles confirms angle congruency, so the triangles are similar. Therefore, the answer is Yes, the triangles are similar. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 5 | Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 5TopicGeometric Shapes DescriptionDetermine if two right triangles, ABC and DEF, are similar. Triangle ABC has angles 90° and 40°, and triangle DEF has angles 90° and 50°. Triangles are similar if all corresponding angles are congruent. Solving reveals congruent corresponding angles. Therefore, the answer is Yes, the triangles are similar. Understanding the concept of geometric shapes is essential for developing geometric reasoning. These examples demonstrate how to analyze and verify similarity between shapes by comparing corresponding sides and angles. The worked-out examples provided help students visualize and understand this key concept. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 6 | Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 6TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are similar. Triangle ABC has a right angle and an angle of 40°, and triangle DEF has a right angle and 45°. Not all corresponding angles are congruent, so the triangles are not similar. Therefore, the answer is No, the triangles aren't similar. Understanding the concept of geometric shapes is essential for developing geometric reasoning. These examples demonstrate how to analyze and verify similarity between shapes by comparing corresponding sides and angles. The worked-out examples provided help students visualize and understand this key concept. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 7 | Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 7TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10 and 12 with a 45° angle, and triangle DEF has sides 5 and 6 with a 45° angle. Triangles are similar if two pairs of sides are proportional, and the included angle is congruent. The side ratio is 2:1, and both included angles are 45°. Therefore, the answer is Yes, the triangles are similar. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 8 | Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 8TopicGeometric Shapes DescriptionDetermine if two triangles, ABC and DEF, are similar. Triangle ABC has sides 10 and 12 with a 45° angle, and triangle DEF has sides 5 and 6 with a 40° angle. Triangles are similar if two pairs of sides are proportional, and the included angles are congruent. Here, the included angles differ (45° vs. 40°), so the triangles are not similar. Therefore, the answer is No, the triangles aren't similar. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 9 | Math Example--Geometric Shapes--Analyzing Similar Shapes--Example 9TopicGeometric Shapes DescriptionDetermine if two right triangles are similar. The triangle on the left has sides 10 and 15 and an included right angle, while the other triangle has sides 5 and 7.5 and an included right angle. Triangles are similar if two pairs of corresponding sides are proportional, and corresponding included angles are congruent. The side ratio is 2:1, and the included right angles are congruent. Therefore, the answer is Yes, the triangles are similar. |
Definition of a Triangle |
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Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 1 | Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 1TopicGeometry DescriptionThis image shows three drumsticks arranged to form a triangle. The drumsticks are placed in a closed triangular shape, with three angles marked in red. This example shows how to construct a triangle using drumsticks. The solution explains that to know it's a triangle, you need to create a closed figure with three sides and note the three interior angles. |
Definition of a Quadrilateral and Geometric Constructions with Triangles |
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Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 10 | Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 10TopicGeometry DescriptionThe image shows red and yellow circular counters arranged in a grid pattern to form a square shape. The solution explains how to arrange the counters in equal numbers per layer to form a square with four clearly defined corners. This example showcases how to construct a square using these red and yellow counters. |
Definition of a Quadrilateral and Geometric Constructions with Triangles |
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Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 2 | Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 2TopicGeometry DescriptionThis image shows a triangle drawn using a crayon. The triangle is closed with three sides, and the three angles are highlighted in red. This example showcases how to draw a triangle using a crayon or pencil. The solution states that to confirm it's a triangle, you need to create a closed figure with three sides and ensure it has three clearly identified angles. |
Definition of a Quadrilateral and Geometric Constructions with Triangles |
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Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 3 | Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 3TopicGeometry DescriptionThis image shows a triangle drawn using paint and a brush. The triangle is closed with three sides, and the three angles are marked in red. This example showcases how to draw a triangle using paint and a brush. The solution explains that to verify it's a triangle, you must create a closed figure with three sides and make sure it has three clearly identified angles. |
Definition of a Quadrilateral and Geometric Constructions with Triangles |
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Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 4 | Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 4TopicGeometry DescriptionThis image shows a triangle formed by arranging stickers in a triangular shape. The stickers form the sides of the triangle, and the angles are highlighted. This example showcases how to draw a triangle using stickers. The solution mentions that you can use stickers to form a closed figure with three sides, ensuring it has three clearly identified angles to confirm it's a triangle. |
Definition of a Quadrilateral and Geometric Constructions with Triangles |
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Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 5 | Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 5TopicGeometry |
Definition of a Quadrilateral and Geometric Constructions with Triangles |
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Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 6 | Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 6TopicGeometry DescriptionThe image features four drumsticks arranged in a square shape. The solution highlights how to verify that the figure is a square by checking that it has four sides and four square corners. This example showcases how to construct a square using four drumsticks. The text describes how to create a closed figure with four sides and check for square corners to confirm it's a square. |
Definition of a Quadrilateral and Geometric Constructions with Triangles |
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Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 7 | Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 7TopicGeometry |
Definition of a Quadrilateral and Geometric Constructions with Triangles |
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Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 8 | Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 8TopicGeometry |
Definition of a Quadrilateral and Geometric Constructions with Triangles |
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Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 9 | Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 9TopicGeometry DescriptionThe image shows a set of colorful stickers with various holiday-themed designs (hearts, snowmen, reindeer, bells, etc.). The stickers are arranged to form a square shape. The solution demonstrates how to use the stickers to create a square by aligning them in a closed figure with four clear corners. This example showcases how to draw a square using these stickers. |
Definition of a Quadrilateral and Geometric Constructions with Triangles |
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Math Example--Geometric Transformation--Translating Triangles--Example 1 | Math Example--Geometric Transformation--Translating Triangles--Example 1TopicTransformations DescriptionA triangle on a grid is translated 4 units to the left. It shows the original triangle ABC and the translated triangle A'B'C'. Example 1: "Draw the triangle that results from the following translation: 4 units to the left." Solution: "Identify one point to translate. Then complete the triangle." |
Applications of Transformations and Applications of Triangles |
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Math Example--Geometric Transformation--Translating Triangles--Example 10 | Math Example--Geometric Transformation--Translating Triangles--Example 10TopicTransformations Description
The triangle is translated 4 units to the left and 2 units up. It displays both the initial and the translated triangles. Example 10: "Draw the triangle that results from the following translation: 4 units to the left, 2 units up." Solution: "Identify one point to translate. Then complete the triangle." |
Applications of Transformations and Applications of Triangles |
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Math Example--Geometric Transformation--Translating Triangles--Example 11 | Math Example--Geometric Transformation--Translating Triangles--Example 11TopicTransformations DescriptionTriangle ABC is translated horizontally to the right by 4 units to form triangle A'B'C'. Example 11: The translation is described as 4 units to the right. In this topic, students explore transformations, focusing specifically on translating triangles. These examples visually demonstrate how shapes move within a coordinate plane, reinforcing understanding of shifts along axes. Translation examples assist in grasping the basic concept of shifting figures without altering their orientation or shape. |
Applications of Transformations and Applications of Triangles |
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Math Example--Geometric Transformation--Translating Triangles--Example 12 | Math Example--Geometric Transformation--Translating Triangles--Example 12TopicTransformations DescriptionTriangle ABC is translated vertically downward by 6 units to form triangle A'B'C'. Example 12: The translation is described as 6 units down. In this topic, students explore transformations, focusing specifically on translating triangles. These examples visually demonstrate how shapes move within a coordinate plane, reinforcing understanding of shifts along axes. Translation examples assist in grasping the basic concept of shifting figures without altering their orientation or shape. |
Applications of Transformations and Applications of Triangles |
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Math Example--Geometric Transformation--Translating Triangles--Example 13 | Math Example--Geometric Transformation--Translating Triangles--Example 13TopicTransformations DescriptionTriangle ABC is translated horizontally to the left by 5 units to form triangle A'B'C'. Example 13: The translation is described as 5 units to the left. In this topic, students explore transformations, focusing specifically on translating triangles. These examples visually demonstrate how shapes move within a coordinate plane, reinforcing understanding of shifts along axes. Translation examples assist in grasping the basic concept of shifting figures without altering their orientation or shape. |
Applications of Transformations and Applications of Triangles |
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Math Example--Geometric Transformation--Translating Triangles--Example 14 | Math Example--Geometric Transformation--Translating Triangles--Example 14TopicTransformations DescriptionTriangle ABC is translated vertically upward by 6 units to form triangle A'B'C'. Example 14: The translation is described as 6 units up. In this topic, students explore transformations, focusing specifically on translating triangles. These examples visually demonstrate how shapes move within a coordinate plane, reinforcing understanding of shifts along axes. Translation examples assist in grasping the basic concept of shifting figures without altering their orientation or shape. |
Applications of Transformations and Applications of Triangles |
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Math Example--Geometric Transformation--Translating Triangles--Example 15 | Math Example--Geometric Transformation--Translating Triangles--Example 15TopicTransformations Description
Triangle ABC is translated diagonally 5 units to the right and 5 units downward to form triangle A'B'C'. Example 15: The translation is described as 5 units to the right and 5 units down. |
Applications of Transformations and Applications of Triangles |
