A right triangle is a triangle in which one of the angles is a right angle, i.e., 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be written as:

c^{2} = a^{2} + b^{2}

- One angle is always 90 degrees.
- The sum of the other two angles is always 90 degrees.
- The sides are related by the Pythagorean theorem.

There are two special types of right triangles: the 45-45-90 triangle and the 30-60-90 triangle. These triangles have specific ratios between their sides that make them useful for solving problems.

In a 45-45-90 triangle, the two legs are congruent, and the length of the hypotenuse is equal to √2 times the length of a leg.

In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg, and the longer leg is √3 times the length of the shorter leg.

Solve the following problems:

- In a right triangle, the lengths of the two legs are 3 and 4. Find the length of the hypotenuse.
- Find the lengths of the sides of a 30-60-90 triangle if the shorter leg has a length of 6.

Understanding the properties of right triangles and the Pythagorean theorem is important for various mathematical and real-world applications. Practice problems and familiarize yourself with special right triangles to strengthen your understanding of the topic.

.Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.