Geometry Unit 11: Similar Polygons

Questions and Answers

What is the cosine ratio?

The side opposite an acute angle over the hypotenuse

The side opposite an acute angle over the side adjacent to the acute angle

The side adjacent to an acute angle over the hypotenuse (correct)

The ratio of two numbers

What is the geometric mean between two positive real numbers a and b?

x such that a/x = x/b

What is the projection of a point on a line?

The point where a perpendicular through the point to the line intersects the line.

What is the projection of a segment on a line?

The portion of a line with endpoints that are the projections of the endpoints of the segment. Signup and view all the answers

What is a proportion?

An equation that states that two ratios are equal. Signup and view all the answers

What is a ratio?

The comparison of two numbers by division. Signup and view all the answers

What defines similar polygons?

Polygons whose vertices can be matched in a one-to-one correspondence so that corresponding angles are equal and corresponding sides are in proportion. Signup and view all the answers

What is the sine ratio?

The side opposite an acute angle over the hypotenuse. Signup and view all the answers

What is the tangent ratio?

The side opposite an acute angle over the side adjacent to the acute angle. Signup and view all the answers

What is Postulate 15?

If the three angles of one triangle are equal to the three angles of another triangle, then the triangles are similar. Signup and view all the answers

What is Theorem 5-1?

If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. Signup and view all the answers

What is Theorem 5-2?

If two sides of one triangle are proportional to two sides of another and the included angles are equal, then the triangles are similar. Signup and view all the answers

What is Theorem 5-3?

If three sides of one triangle are proportional to three sides of another, then the triangles are similar. Signup and view all the answers

What is Theorem 5-4?

Similarity of polygons is reflexive, symmetric, and transitive. Signup and view all the answers

What is Theorem 5-5?

If two polygons are similar, then the ratio of their perimeters is the same as the ratio of any pair of corresponding sides. Signup and view all the answers

What is Theorem 5-6?

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the two sides proportionally. Signup and view all the answers

What is Theorem 5-7?

If a ray bisects an angle of a triangle, then it divides the opposite side into segments with lengths proportional to the lengths of the other two sides of the triangle. Signup and view all the answers

What is Theorem 5-8?

If two triangles are similar, then the lengths of corresponding altitudes have the same ratio as the lengths of any pair of corresponding sides. Signup and view all the answers

What is Theorem 5-9?

If the altitude to the hypotenuse of a right triangle is drawn, then the two triangles formed are similar to each other and similar to the given triangle. Signup and view all the answers

What is Theorem 5-10?

In a right triangle, the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse. Signup and view all the answers

What is Theorem 5-11?

If the sum of the squares of two sides of a triangle equals the square of the third side, then the triangle is a right triangle. Signup and view all the answers

What is Theorem 5-12?

In a 30°-60°-90° triangle, the hypotenuse is twice the short leg, and the longer leg is $ ext{√3}$ times the short leg. Signup and view all the answers

What is Theorem 5-13?

If each acute angle of a triangle is 45°, then the measure of the hypotenuse is $ ext{√2}$ times the leg. Signup and view all the answers

Study Notes

Trigonometric Ratios

Cosine Ratio: In a right triangle, defined as the length of the side adjacent to an acute angle divided by the hypotenuse.
Sine Ratio: In a right triangle, the ratio of the length of the side opposite an acute angle to the hypotenuse.
Tangent Ratio: The ratio of the length of the side opposite an acute angle to the length of the side adjacent to that angle.

Means and Proportions

Geometric Mean: The positive real number x such that the ratio a/x equals x/b for positive real numbers a and b.
Proportion: An equation that asserts two ratios are equal.
Ratio: Comparison of two numbers via division, where the quotient is termed the ratio.

Similar Polygons and Triangles

Similar Polygons: Polygons with corresponding angles equal and corresponding sides in proportion.
Postulate 15: Triangles are similar if all three angles are equal (AAA).
Theorem 5-1: If two angles of one triangle equal two angles of another triangle, they are similar.
Theorem 5-2: If two sides of one triangle are proportional to two sides of another and the included angles are equal, the triangles are similar.
Theorem 5-3: Similarity is established if three sides of one triangle are proportional to three sides of another triangle.

Properties of Similarity

Theorem 5-4: Similarity of polygons is reflexive, symmetric, and transitive.
Theorem 5-5: For similar polygons, the ratio of their perimeters matches the ratio of corresponding sides.

Triangle Properties and Theorems

Theorem 5-6: A line parallel to one side of a triangle divides the other two sides proportionally.
Theorem 5-7: An angle bisector in a triangle divides the opposite side into segments proportional to the other two sides.
Theorem 5-8: The lengths of corresponding altitudes in similar triangles share the same ratio as corresponding sides.
Theorem 5-9: In a right triangle, drawing the altitude to the hypotenuse forms two triangles that are similar to each other and to the original triangle.
Theorem 5-10: The Pythagorean Theorem states that the sum of the squares of the legs equals the square of the hypotenuse in a right triangle.
Theorem 5-11: A triangle is a right triangle if the sum of the squares of two sides equals the square of the third side.

Special Right Triangles

Theorem 5-12: In a 30°-60°-90° triangle, the hypotenuse is double the length of the short leg; the longer leg is √3 times the short leg.
Theorem 5-13: In a 45°-45° right triangle, the hypotenuse measures √2 times the length of each leg.

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Test your knowledge on similar polygons with this quiz covering important concepts such as cosine ratios and geometric means. Dive into the definitions and relationships that define these shapes in geometry. Perfect for students looking to understand the properties of polygons better!