Exploring Analyic Geometry with Mathematica®

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Angle Inscribed in a Semicircle

rtangcir.html

Exploration

Show that an angle inscribed in a semicircle is a right angle.

Approach

Find the parametric coordinates of the points that define the angle and use the Pythagorean Theorem to show they form a right angle.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Create a circle at the origin.

Clear[r];
C1=Circle2D[{0,0},r];

Construct the points on the semicircle.  "rtangcir_1.gif" and "rtangcir_2.gif" are the end points of the semicircle, "rtangcir_3.gif" is the (right) angle vertex.

Clear[t];
P1=C1[0];
P2=C1[t];
P3=C1[Pi];

Apply the Pythagorean Theorem. First compute "rtangcir_4.gif" and then show it is equal to "rtangcir_5.gif" and independent of the parameter value of the vertex point (it turns out that it is a function of the circle's radius only).

{Distance2D[P1,P2]^2+Distance2D[P2,P3]^2,
Distance2D[P1,P3]^2} //Simplify

"rtangcir_6.gif"


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