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NY Regents Exam - Geometry: Help and Review10 chapters | 127 lessons

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Lesson Transcript

Instructor:
*Kimberlee Davison*

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

In this lesson, you will learn about slope triangles, a kind of triangle that helps you easily find the slope of a line or line segment. Often, these are imaginary triangles sketched between two points.

A **slope triangle** is a visual tool that helps you find the slope of a line. By '**slope**,' we mean steepness.

Imagine that you are flying a plane and have ascended (vertically) 1,000 feet. When you are 1,000 feet above the ground, you have traveled forward (horizontally) 3,000 feet. With those two measurements, you can figure out how steep your ascent was. The higher the plane rises while it travels forward a certain amount, the steeper the incline.

If you look at the picture of the airplane's ascent, you'll see a triangle. This triangle is imaginary, of course. It is a visual tool, a 'slope triangle,' that helps you calculate how steep the ascent was. In other words, it helps you figure out the slope of the line connecting the plane's starting point to its current position.

One formula for slope you may have seen looks like this:

*Slope = rise/run*

The rise is the vertical distance on the triangle. The run is the horizontal distance. For our airplane, we get:

Slope = 1,000 feet/3,000 feet = 1/3

In other words, the airplane rises 1 foot vertically every time it travels 3 feet along the ground.

If a plane is 2,000 feet above the ground after it has traveled 3,000 feet forward, then its ascent is steeper. For the same horizontal distance traveled, it has reached a greater height. In this case, the slope triangle is less flat. The horizontal leg of the triangle is longer.

Slope = rise/run = 2,000 ft/3,000 ft = 2/3

The slope of the first airplane's ascent, 1/3, is less than the slope of the second airplane, 2/3. The second airplane travels upward more steeply.

Sometimes you may want to find the slope of a line, a line segment, or an imaginary line segment between two points. In the picture, the red dot is at the point (1,2). The blue dot is at the point (2,4). You can draw in a slope triangle by connecting the two points and creating the horizontal and vertical legs of a triangle.

The rise is the length of the vertical leg, 2 units. The run is the length of the horizontal leg, 1 unit. The slope is found by dividing:

Slope = rise/run = 2/1 = 2

Slope triangles work when a line slants downward as well. In the next picture, the blue dot (2,4) is higher than the red dot (5,2). In this case, the rise is -2 because you are moving downward from left to right. So, rise/run = -2/3. If the dots represented an airplane's location, then the airplane would be descending, or moving closer to the earth.

A **slope triangle** is an imaginary triangle that helps you find the slope of a line or a line segment. The hypotenuse of the triangle (the diagonal) is the line you are interested in finding the slope of. The two 'legs' of the triangle are the 'rise' and 'run' used in the slope formula. *Slope = rise/run*.

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29 in chapter 8 of the course:

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NY Regents Exam - Geometry: Help and Review10 chapters | 127 lessons

- Triangles: Definition and Properties 4:30
- Area of Triangles and Rectangles 5:43
- Classifying Triangles by Angles and Sides 5:44
- Perimeter of Triangles and Rectangles 8:54
- Interior and Exterior Angles of Triangles: Definition & Examples 5:25
- How to Identify Similar Triangles 7:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:16
- Applications of Similar Triangles 6:23
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
- Converse of a Statement: Explanation and Example 5:09
- Median, Altitude, and Angle Bisectors of a Triangle 4:50
- Properties of Concurrent Lines in a Triangle 6:17
- Angles and Triangles: Practice Problems 7:43
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Constructing Triangles: Types of Geometric Construction 5:59
- Constructing the Median of a Triangle 4:47
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
- The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples 5:50
- The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples 6:19
- AA Similarity Postulate & Theorem 5:56
- Circumcenter: Definition, Formula & Construction
- Half-Angle: Formulas & Proof
- Intercepted Arc: Definition & Formula 2:47
- Percent Decrease: Formula & Calculation
- Similar Triangles: Definition, Formula & Properties 6:43
- Slope Triangle: Definition & Concept 3:14
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