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Study Notes 017:

By William E. Steinman:

Geometry 10 Circles and Pythagoras:

Post Date:

The length of a tangent from a point to a circle is the length of the line segment from the point to the point of tangency.

A tangent is perpendicular to the radius drawn to the point of contact.

Logically, a line will pass through the center of a circle if it is perpendicular to a tangent at the point of contact.

Tangents to a circle from an outside point are congruent.

The line segment from the center of a circle to an outside point bisects the angle between the tangents.

The line of centers of two circles is the line joining their centers.

Circles can be tangent externally or internally.

Of course, circles can also overlap.

A central angle is an angle having its vertex at the center of a circle.

Cleary then, a central angle has the same number of degrees as the arc it intercepts.

We could also say, a central angle is measured by its intercepted arc.

An inscribed angle is an angle whose vertex is on the circle and whose sides are chords.

While a central angle is measured by its intercepted arc, a inscribed angle is measured by one-half it intercepted arc.

To review some items from algebra or math:

Ratios are used to compare quantities by division. So the ratio of two quantities is the first divided by the second. Clearly, there are no units of measure in ratios. They are abstract. Therefore, the quantities involved in a ration must have the same units.

A proportion is an equality of two ratios.

For example 4:8 = 8:16 is a proportion.

A clearer way to state that is to say 4/8 is proportional to 8/16.

Of course, it’s simple math.

It is the basis for solving problems for one unknown with algebraic equations.

If two line segments are divided proportionally, the corresponding new segments are in proportion.

If a line is parallel to one side of a triangle, it divides the other two sides proportionally.

A bisector of an angle or a triangle divides the opposite side into segments that are proportional to the adjacent sides.

Similar polygons are polygons whose corresponding angles are congruent and whose sides are in proportion. That is, they have the same shape but not necessarily the same size.

Tow triangles are similar if two angles of one triangle are congruent to two angles of the other.

Corresponding sides of similar triangles are n proportion.

Corresponding segments of similar triangles are in proportion.

Corresponding segments of similar polygons are in proportion.

If two chords intersect within a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

Therefore AC x CE = DC x CB.

Pythagorean Theorem:

Pythagoras worked this out about two centuries before Euclid was born.

In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Therefore, in the above triangle H2 = X2 + Y2.

The classic of this, which is very handy for laying out square fences and building, is the 3, 4, 5 right triangle.

Where X is equal to 3 or some multiple thereof, Y is equal to 4 or the same multiple thereof, and H is equal to 5 or the same multiple thereof.

This gives us 32 + 42 = 52 or 9 + 16 = 25. Elementary my dear Watson.

Certain concepts are useful for solving some problems concerning plane figures.

For example:

A rectangle or square can be divided into two right triangles.

An equilateral triangle can be divided into two right triangles.