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- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/06%3A_Triangles_and_Circles/6.03%3A_Similar_TrianglesThis section focuses on similar triangles, highlighting their definition, congruence, and applications. It introduces the concept of similarity, demonstrates how to identify similar triangles through ...This section focuses on similar triangles, highlighting their definition, congruence, and applications. It introduces the concept of similarity, demonstrates how to identify similar triangles through examples, and explores their properties, including proportional sides and angles. Key topics include using proportions to solve problems involving similar triangles, understanding similar right triangles, and dealing with overlapping triangles.
- https://math.libretexts.org/Bookshelves/Geometry/Elementary_College_Geometry_(Africk)/06%3A_Area_and_Perimeter/6.02%3A_The_Area_of_a_ParallelogramThe base may be any side of the parallelogram, though it is usually chosen to be the side on which the parallelogram appears to be resting. The height is a line drawn perpendicular to the base from th...The base may be any side of the parallelogram, though it is usually chosen to be the side on which the parallelogram appears to be resting. The height is a line drawn perpendicular to the base from the opposite side.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/01%3A_Triangles_and_Circles/1.03%3A_Similar_TrianglesThis section focuses on similar triangles, highlighting their definition, congruence, and applications. It introduces the concept of similarity, demonstrates how to identify similar triangles through ...This section focuses on similar triangles, highlighting their definition, congruence, and applications. It introduces the concept of similarity, demonstrates how to identify similar triangles through examples, and explores their properties, including proportional sides and angles. Key topics include using proportions to solve problems involving similar triangles, understanding similar right triangles, and dealing with overlapping triangles.
- https://math.libretexts.org/Courses/Coalinga_College/Math_for_Educators_(MATH_010A_and_010B_CID120)/13%3A_Area_Pythagorean_Theorem_and_Volume/13.02%3A_Parallelogramsw∘=115∘ since the opposite angles of a parallelogram are equal. \(x^{\circ} = 180^{\circ} -(w^{\circ} + 30^{\circ}) = 180^{\circ} - (115^{\circ} + 30^{\circ}) = 180^{\circ} - 145^{...w∘=115∘ since the opposite angles of a parallelogram are equal. x∘=180∘−(w∘+30∘)=180∘−(115∘+30∘)=180∘−145∘=35∘, because the sum of the angles of △ABC is 180∘, y∘=30∘ and x∘=x∘=35∘ because they are alternate interior angles of parallel lines.
- https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Technical_Math_1A_(NWTC)/07%3A_Geometry/7.02%3A_Area_of_Polygons_and_CirclesWe have seen that the perimeter of a polygon is the distance around the outside. Perimeter is a length, which is one-dimensional, and so it is measured in linear units (feet, centimeters, miles, etc.)...We have seen that the perimeter of a polygon is the distance around the outside. Perimeter is a length, which is one-dimensional, and so it is measured in linear units (feet, centimeters, miles, etc.). The area of a polygon is the amount of two-dimensional space inside the polygon, and it is measured in square units: square feet, square centimeters, square miles, etc.
- https://math.libretexts.org/Bookshelves/Geometry/Elementary_College_Geometry_(Africk)/03%3A_Quadrilaterals/3.01%3A_Parallelogramsw∘=115∘ since the opposite angles of a parallelogram are equal. \(x^{\circ} = 180^{\circ} -(w^{\circ} + 30^{\circ}) = 180^{\circ} - (115^{\circ} + 30^{\circ}) = 180^{\circ} - 145^{...w∘=115∘ since the opposite angles of a parallelogram are equal. x∘=180∘−(w∘+30∘)=180∘−(115∘+30∘)=180∘−145∘=35∘, because the sum of the angles of △ABC is 180∘, y∘=30∘ and x∘=x∘=35∘ because they are alternate interior angles of parallel lines.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.02%3A_Dot_and_Cross_Product/4.2E%3A_Exercises_for_Section_4.2This page offers a series of exercises in vector operations and geometry, including dot products, Cauchy-Schwarz inequality, projections, and the cross product. It covers tasks such as calculating ang...This page offers a series of exercises in vector operations and geometry, including dot products, Cauchy-Schwarz inequality, projections, and the cross product. It covers tasks such as calculating angles, distances from points to lines, and areas of triangles. Key concepts include relationships in vector algebra, properties of matrix multiplication, and geometric significance of vector computations, such as coplanarity indicated by the zero box product and the volume of parallelepipeds.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.04%3A_Determinants_and_Geometry/3.4E%3A_Exercises_for_Section_3.4This page presents a collection of geometry and linear algebra exercises that involve calculating areas and volumes of shapes derived from vectors. It includes tasks on finding the areas of parallelog...This page presents a collection of geometry and linear algebra exercises that involve calculating areas and volumes of shapes derived from vectors. It includes tasks on finding the areas of parallelograms and triangles, determining conditions for zero areas or volumes, and exploring determinants' properties in both 2D and 3D. The content suggests theoretical proofs regarding skew-symmetric matrices and vector dependencies and includes specific variable values when needed.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.01%3A_Review_of_Vectors/4.1E%3A_Exercises_for_Section_4.1This page covers exercises on vector operations, including linear combinations, vector properties, and geometric interpretations. It starts with calculations involving vectors and progresses to relati...This page covers exercises on vector operations, including linear combinations, vector properties, and geometric interpretations. It starts with calculations involving vectors and progresses to relationships between vector sums and geometric shapes like parallelograms. It requires proofs of unique zero vectors and properties of vector addition, culminating in insights into linear combinations and geometric implications.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.04%3A_Determinants_and_GeometryIn this section we give a geometric interpretation of determinants, in terms of volumes. This will shed light on the reason behind three of the four defining properties of the determinant. It is also ...In this section we give a geometric interpretation of determinants, in terms of volumes. This will shed light on the reason behind three of the four defining properties of the determinant. It is also a crucial ingredient in the change-of-variables formula in multivariable calculus.
