Exploring the Tangent Ratio

What does "Opposite" and "Adjacent" mean?

Each angle in a triangle has an opposite and adjacent side. In this example, the opposite side of angle Q is the line PR and the adjacent side is line RQ. For angle R, the opposite side is the line PQ and the adjacent side is the line RQ.

What is the length of the opposite side and what is the length of the adjacent side of the angle alpha shown above?

What is the "tangent ratio"?

This relationship between the opposite side and the adjacent side is called "tangent ratio". Recall that ratios are represented as fractions, so we write the tangent ratio of alpha here as 8/15. We call this number 8/15 (or .533 as a decimal) the "tangent of alpha".

STUDENTS:

Perform these 3 tasks to setup our exploration for today:[br](1) Use the Polygon tool [icon]https://www.geogebra.org/images/ggb/toolbar/mode_polygon.png[/icon] to create a triangle that contains one 90° degree angle and two other angles of your choosing. Display side lengths and angle measurements on each side and angle in each triangle (If you need help using GeoGebra to do this step, watch the video shown below called "How to Draw Triangles in GeoGebra")[br][br](2) Repeat step (1) twice more.[br][br](3) Change the side lengths of 2 of the triangles so that all sides on all the triangles are different length.

What are [b](a)[/b] the [u]lengths[/u] of the opposite and adjacent sides of one of the acute angles on the first triangle you drew, and [b](b)[/b] what is their [u]ratio[/u]?

What are [b](a)[/b] the [u]lengths[/u] of the opposite and adjacent sides of one the acute angles on the second triangle you drew, and [b](b)[/b] what is their [u]ratio[/u]?

What are [b](a)[/b] the [u]lengths[/u] of the opposite and adjacent sides of one the acute angles on the third triangle you drew, and [b](b)[/b] what is their [u]ratio[/u]?

What do you notice about the ratios on all 3 triangles?

What is the length of the opposite side of the angle alpha [math]\alpha[/math]?

6 [math]\sqrt{52}[/math] 4

What is the length of the opposite side of the angle alpha [math]\beta[/math]?

4 6 [math]\sqrt{52}[/math]

What is the tangent ratio of the angle alpha [math]\alpha[/math]?

[math]\frac{\sqrt{52}}{6}[/math] [math]\frac{4}{\sqrt{52}}[/math] [math]\frac{\sqrt{52}}{4}[/math] [math]\frac{4}{6}[/math] [math]\frac{6}{\sqrt{52}}[/math] [math]\frac{6}{4}[/math]

[math][/math]What is the tangent ratio of the angle beta?

[math]\frac{\sqrt{52}}{4}[/math] [math]\frac{6}{\sqrt{52}}[/math] [math]\frac{4}{\sqrt{52}}[/math] [math]\frac{4}{6}[/math] [math]\frac{\sqrt{52}}{6}[/math] [math]\frac{6}{4}[/math]

Exploring the Tangent Ratio

What does "Opposite" and "Adjacent" mean?

What is the "tangent ratio"?

STUDENTS:

Information: Exploring the Tangent Ratio