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To discover the tangent ratio in a
right triangle using similar triangles.
The Tangent Ratio
Introduction
Since the ancient Egyptians, surveyors have used triangles to plot land, and lay out foundations for structures. Surveyors need accuracy, but may not be able to collect all the necessary measurements because of geographical features of the land. Knowing an angle measurement and one side length of a right triangle is enough to find all the other important measurements in the triangle. For example suppose a surveyor is measuring the distance across a river so they can build a bridge as shown below.
SHAPE \* MERGEFORMAT
A surveyor at point A measures the angle shown in the triangle at point A. The surveyors also know the distance from point A to the point directly across the river from point B. So with that information can they find the distance across the river?
Construction & Exploration
Part I Construct right triangle
Construct a horizontal line near the bottom of the screen.
Place two points on the line. Label the points A and C.
Construct a perpendicular through point C perpendicular
the line AC.
Construct a line through point A so that it intersects the
perpendicular line you constructed in step 3 above. Be sure
the other point that defines the line does not lie on the
perpendicular line.
Find the point of intersection of the line through A and the
perpendicular line constructed in step 3. Label the point B.
Part II. Measuring angles and sides.
Measure EMBED Equation.3 .
Measure BC and AC.
Calculate the ratio of EMBED Equation.3 .
Questions and Conjectures
Move point C, observe the measurement of EMBED Equation.3 . The new triangles formed by moving point C are considered to be _______________.
Observe the ratio as you move point C. What occurs with regard to the ratio?
Change EMBED Equation.3 to a different measurement less than 90. Observe the ratio while moving point C.
Conjecture about the ratio of the two sides in similar right triangles.
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