Question
Unit 6 Right Triangles and the Pythagorean Theorem
Vocabulary
Key Terms
Meaning/definition/Picture
Example
U6 L1
Acute angle
Obtuse angle
U6 L1
Right Triangle
U6 L2
Triangle Angle Sum Theorem
U6 L2
Right triangle side length relationship
U6 L4
Hypotenuse
Legs
U6 L4
Pythagorean Theorem
U6 L4
Proof of the Pythagorean Theorem
U6 L5
How do you know which side is the hypotenuse?
U6 L5
What is the order of operations for solving a2+b2=c2
U6 L6
Finding the distance between 2 points in the coordinate plane
Distance Formula:
Pythagorean Theorem:
U6 L7
3 Dimensional Prisms and Pyramids
Prism
Pyramids
U6 L7
Using PT in 3D
Vocabulary
Key Terms
Meaning/definition/Picture
Example
U6 L1
Acute angle
Obtuse angle
U6 L1
Right Triangle
U6 L2
Triangle Angle Sum Theorem
U6 L2
Right triangle side length relationship
U6 L4
Hypotenuse
Legs
U6 L4
Pythagorean Theorem
U6 L4
Proof of the Pythagorean Theorem
U6 L5
How do you know which side is the hypotenuse?
U6 L5
What is the order of operations for solving a2+b2=c2
U6 L6
Finding the distance between 2 points in the coordinate plane
Distance Formula:
Pythagorean Theorem:
U6 L7
3 Dimensional Prisms and Pyramids
Prism
Pyramids
U6 L7
Using PT in 3D
Answers
GPT-4o mini
Sure! Here's the vocabulary filled out based on your unit topic "Right Triangles and the Pythagorean Theorem":
### Vocabulary
| Key Terms | Meaning/Definition/Picture | Example |
|----------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------|
| **U6 L1 <br> Acute angle** | An angle that measures less than 90 degrees.  | The angle in triangle ABC where angle A = 45°. |
| **U6 L1 <br> Obtuse angle** | An angle that measures more than 90 degrees but less than 180 degrees.  | The angle in triangle DEF where angle D = 120°. |
| **U6 L1 <br> Right Triangle** | A triangle that has one angle measuring exactly 90 degrees.  | A triangle with vertices A (0,0), B (3,0), and C (3,4) is a right triangle. |
| **U6 L2 <br> Triangle Angle Sum Theorem** | The theorem stating that the sum of the angles in any triangle is always 180 degrees.  | In triangle XYZ, angle X = 60°, angle Y = 70°, and angle Z = 50°; 60° + 70° + 50° = 180°. |
| **U6 L2 <br> Right triangle side length relationship** | In a right triangle, the relationship between the lengths of the sides can be expressed using the Pythagorean Theorem. | For a right triangle with legs of lengths 3 and 4, the hypotenuse (c) is 5, since 3² + 4² = 5². |
| **U6 L4 <br> Hypotenuse** | The longest side of a right triangle, opposite the right angle.  | In triangle ABC, if angle C is the right angle, then side AB is the hypotenuse. |
| **U6 L4 <br> Legs** | The two shorter sides of a right triangle that form the right angle.  | In triangle ABC, if angle C is the right angle, then sides AC and BC are the legs. |
| **U6 L4 <br> Pythagorean Theorem** | A formula used to find the lengths of the sides of a right triangle: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse and \( a \) and \( b \) are the legs. | If one leg is 3 and the other leg is 4, then according to the Pythagorean Theorem, \( 3^2 + 4^2 = 5^2 \). |
| **U6 L4 <br> Proof of the Pythagorean Theorem** | A mathematical demonstration showing that the Pythagorean theorem is true, often using geometric constructions like squares on each side of a triangle. | One common proof involves constructing squares on each side and demonstrating area equivalence. |
| **U6 L5 <br> How do you know which side is the hypotenuse?** | The hypotenuse is always opposite the right angle in a right triangle and is the longest side of the triangle. | In triangle XYZ, if angle Z is 90°, then side XY is the hypotenuse. |
| **U6 L5 <br> What is the order of operations for solving \( a^2 + b^2 = c^2 \)** | Calculate the squares of the lengths of the legs (a and b), then sum them to find \( c^2 \) and lastly take the square root to find the length of the hypotenuse (c). | Given legs a = 3, b = 4: Calculate \( 3^2 + 4^2 = 9 + 16 = 25 \), thus \( c = \sqrt{25} = 5 \). |
| **U6 L6 <br> Finding the distance between 2 points in the coordinate plane** | The distance between two points (x₁, y₁) and (x₂, y₂) can be found using the Distance Formula: \( d = \sqrt{(x₂ - x₁)^2 + (y₂ - y₁)^2} \). | The distance between points (1, 2) and (4, 6) is \( \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = 5 \). |
| **Distance Formula:** | Formula for calculating the distance between two points in a Cartesian plane. | \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) |
| **Pythagorean Theorem:** | Formula to relate the lengths of the sides of a right triangle: \( a^2 + b^2 = c^2 \). | As previously stated, it's a key theorem for right triangles. |
| **U6 L7 <br> 3 Dimensional Prisms and Pyramids** | Solid shapes that have three dimensions; prisms have two parallel faces, while pyramids have a polygon base and triangular faces that converge at a vertex. | A cube is a prism; a pyramid with a square base is a square pyramid. |
| **Prism** | A 3D shape with two congruent bases connected by rectangular faces.  | A rectangular prism has bases that are rectangles. |
| **Pyramids** | A 3D shape with a polygonal base and triangular faces that meet at a single point (the apex).  | A triangular pyramid has a triangular base and three triangular faces. |
| **U6 L7 <br> Using PT in 3D** | The Pythagorean theorem can be extended to solve problems in three-dimensional space by relating distances within a prism or a pyramid. | To find the diagonal length of a rectangular prism, the formula becomes \( d^2 = l^2 + w^2 + h^2 \). |
This structure includes the key terms and their definitions, as well as examples to illustrate each concept. Let me know if you need further details or adjustments!
### Vocabulary
| Key Terms | Meaning/Definition/Picture | Example |
|----------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------|
| **U6 L1 <br> Acute angle** | An angle that measures less than 90 degrees.  | The angle in triangle ABC where angle A = 45°. |
| **U6 L1 <br> Obtuse angle** | An angle that measures more than 90 degrees but less than 180 degrees.  | The angle in triangle DEF where angle D = 120°. |
| **U6 L1 <br> Right Triangle** | A triangle that has one angle measuring exactly 90 degrees.  | A triangle with vertices A (0,0), B (3,0), and C (3,4) is a right triangle. |
| **U6 L2 <br> Triangle Angle Sum Theorem** | The theorem stating that the sum of the angles in any triangle is always 180 degrees.  | In triangle XYZ, angle X = 60°, angle Y = 70°, and angle Z = 50°; 60° + 70° + 50° = 180°. |
| **U6 L2 <br> Right triangle side length relationship** | In a right triangle, the relationship between the lengths of the sides can be expressed using the Pythagorean Theorem. | For a right triangle with legs of lengths 3 and 4, the hypotenuse (c) is 5, since 3² + 4² = 5². |
| **U6 L4 <br> Hypotenuse** | The longest side of a right triangle, opposite the right angle.  | In triangle ABC, if angle C is the right angle, then side AB is the hypotenuse. |
| **U6 L4 <br> Legs** | The two shorter sides of a right triangle that form the right angle.  | In triangle ABC, if angle C is the right angle, then sides AC and BC are the legs. |
| **U6 L4 <br> Pythagorean Theorem** | A formula used to find the lengths of the sides of a right triangle: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse and \( a \) and \( b \) are the legs. | If one leg is 3 and the other leg is 4, then according to the Pythagorean Theorem, \( 3^2 + 4^2 = 5^2 \). |
| **U6 L4 <br> Proof of the Pythagorean Theorem** | A mathematical demonstration showing that the Pythagorean theorem is true, often using geometric constructions like squares on each side of a triangle. | One common proof involves constructing squares on each side and demonstrating area equivalence. |
| **U6 L5 <br> How do you know which side is the hypotenuse?** | The hypotenuse is always opposite the right angle in a right triangle and is the longest side of the triangle. | In triangle XYZ, if angle Z is 90°, then side XY is the hypotenuse. |
| **U6 L5 <br> What is the order of operations for solving \( a^2 + b^2 = c^2 \)** | Calculate the squares of the lengths of the legs (a and b), then sum them to find \( c^2 \) and lastly take the square root to find the length of the hypotenuse (c). | Given legs a = 3, b = 4: Calculate \( 3^2 + 4^2 = 9 + 16 = 25 \), thus \( c = \sqrt{25} = 5 \). |
| **U6 L6 <br> Finding the distance between 2 points in the coordinate plane** | The distance between two points (x₁, y₁) and (x₂, y₂) can be found using the Distance Formula: \( d = \sqrt{(x₂ - x₁)^2 + (y₂ - y₁)^2} \). | The distance between points (1, 2) and (4, 6) is \( \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = 5 \). |
| **Distance Formula:** | Formula for calculating the distance between two points in a Cartesian plane. | \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) |
| **Pythagorean Theorem:** | Formula to relate the lengths of the sides of a right triangle: \( a^2 + b^2 = c^2 \). | As previously stated, it's a key theorem for right triangles. |
| **U6 L7 <br> 3 Dimensional Prisms and Pyramids** | Solid shapes that have three dimensions; prisms have two parallel faces, while pyramids have a polygon base and triangular faces that converge at a vertex. | A cube is a prism; a pyramid with a square base is a square pyramid. |
| **Prism** | A 3D shape with two congruent bases connected by rectangular faces.  | A rectangular prism has bases that are rectangles. |
| **Pyramids** | A 3D shape with a polygonal base and triangular faces that meet at a single point (the apex).  | A triangular pyramid has a triangular base and three triangular faces. |
| **U6 L7 <br> Using PT in 3D** | The Pythagorean theorem can be extended to solve problems in three-dimensional space by relating distances within a prism or a pyramid. | To find the diagonal length of a rectangular prism, the formula becomes \( d^2 = l^2 + w^2 + h^2 \). |
This structure includes the key terms and their definitions, as well as examples to illustrate each concept. Let me know if you need further details or adjustments!