Geometry Chapter: Fractions and Triangle Properties

Questions and Answers

What are the extremes of two fractions?

The numerator of the first fraction (correct)

The denominator of the second fraction (correct)

The numerator of the second fraction

The denominator of the first fraction

What are the means of two fractions?

The denominator of the first fraction (correct)

The denominator of the second fraction

The numerator of the first fraction

The numerator of the second fraction (correct)

What is Geometric Mean?

The positive square root of the product of two numbers.

What is altitude?

Segment from the vertex perpendicular to the side opposite or the line containing the side opposite. Signup and view all the answers

What is the Altitude of Right Triangle Theorem?

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

What is the Geometric Mean (Altitude) Theorem?

The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments and is the geometric mean between the lengths of these two segments. Signup and view all the answers

What is the Geometric Mean (Leg) Theorem?

The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments, and the length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Signup and view all the answers

Define hypotenuse.

The longest side of a right triangle and opposite of the right angle. Signup and view all the answers

What is the Pythagorean Theorem?

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

What is the Converse Pythagorean Theorem?

If the sum of the square of the two smaller sides of a triangle are equal to the square of the larger side, then the triangle is a right triangle. Signup and view all the answers

Define a Pythagorean triple.

A set of three non-zero whole numbers, a, b, and c, so that a² + b² = c². Signup and view all the answers

What are the Pythagorean Inequality Theorems?

<ol> <li>If c² > a² + b² then the triangle is obtuse. 2. If c² < a² + b² then the triangle is acute. 3. If c² = a² + b² then the triangle is a right triangle.</li> </ol> Signup and view all the answers

What is the 45-45-90 Triangle Theorem?

In a 45-45-90 triangle, the legs are congruent and the length of the hypotenuse is the length of the legs times √2. Signup and view all the answers

What is the 30-60-90 Triangle Theorem?

In a 30-60-90 triangle, the length of the hypotenuse is two times the length of the short leg, and the long leg is √3 times the short leg. Signup and view all the answers

What are the abbreviations to remember Trigonometric Ratios?

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You use inverse trigonometric ratios to find...

angles. Signup and view all the answers

What do you need to know to solve a right triangle?

You need to know the measurements of two sides of the right triangle or one side and one acute angle. Signup and view all the answers

Define Angle of Elevation.

Always measured from the ground up, it's an upward angle from a horizontal line. Signup and view all the answers

Define Angle of Depression.

Angle formed by the horizontal and the observer's line of sight to an object below. Signup and view all the answers

The Angle of Elevation is ________________ to the Angle of Depression.

equal Signup and view all the answers

The Law of Sines can be used to find the side lengths and angle measures for ________ triangle.

any Signup and view all the answers

What is the Law of Sines?

If triangle ABC has lengths a, b, and c, then sinA/a = sinB/b = sinC/c. Signup and view all the answers

When the Law of Sines cannot be used to solve a triangle, the _____________________________________ may apply.

Law of Cosines. Signup and view all the answers

What is the Law of Cosines?

If triangle ABC has lengths a, b, and c, then a² = b² + c² - 2bc cosA, b² = a² + c² - 2ac cosB, c² = a² + b² - 2ab cosC. Signup and view all the answers

When you solve a triangle, you...

find the length of every side and the degree of every angle. Signup and view all the answers

When do you use trigonometric ratios (SOH CAH TOA)?

When the triangle has a right angle. Signup and view all the answers

When do you use the Law of Sines?

When you have a matching pair on a non-right triangle. Signup and view all the answers

When do you use the Law of Cosines?

When you have a non-right triangle and you do not have a matching pair. Signup and view all the answers

When do you use geometric mean?

When you are given one large triangle and two component triangles within that large triangle. Signup and view all the answers

When do you use the Pythagorean Theorem?

When you are finding the side of a right triangle. Signup and view all the answers

If two legs are congruent, then...

their opposite base angles are congruent. Signup and view all the answers

When solving any mathematical problem, always make sure the measurements are the ___________________.

same Signup and view all the answers

What are the proportions for a 45-45-90 and 30-60-90 degree triangle?

45-45-90: 1 go on legs, and √2 goes on hypotenuse. 30-60-90: 1 goes opposite 30°, 2 goes opposite 90°, and √3 goes opposite 60°. Signup and view all the answers

What does it mean to be a matching pair?

A matching pair has a side and an opposite angle that both have their appropriate numerical measurements. Signup and view all the answers

When is a hypotenuse present?

In a right triangle. Signup and view all the answers

Study Notes

Fractions and Means

Extremes of two fractions: numerator of the first fraction and denominator of the second fraction.
Means of two fractions: denominator of the first fraction and numerator of the second fraction.

Geometric Mean

Defined as the positive square root of the product of two numbers.
If a/x = x/b, then x² = ab, implying x = √(ab).

Triangle Properties

Altitude: a segment from the vertex of a triangle perpendicular to the opposite side.
Altitude of Right Triangle Theorem: altitude to the hypotenuse creates two triangles that are similar to the original triangle.

Geometric Mean Theorems

Geometric Mean (Altitude) Theorem: altitude to the hypotenuse is the geometric mean of the two segments it creates.
Geometric Mean (Leg) Theorem: length of a leg is the geometric mean of the hypotenuse and the adjacent segment of the hypotenuse.

Key Triangle Terminology

Hypotenuse: the longest side of a right triangle, opposite the right angle.
Pythagorean Theorem: in a right triangle, a² + b² = c², where c is the hypotenuse.
Converse Pythagorean Theorem: if a² + b² = c², the triangle is a right triangle.
Pythagorean triple: a set of three non-zero whole numbers, (a, b, c), satisfying a² + b² = c².

Pythagorean Inequality Theorems

If c² > a² + b², the triangle is obtuse.
If c² < a² + b², the triangle is acute.
If c² = a² + b², the triangle is a right triangle.

Special Triangle Theorems

45-45-90 Triangle Theorem: legs are congruent, hypotenuse = leg length × √2.
30-60-90 Triangle Theorem: hypotenuse = 2 × short leg; long leg = short leg × √3.

Trigonometric Ratios

Use trigonometric ratios (SOH CAH TOA) for right triangles only involving acute angles.
Inverse trigonometric ratios used to find angles, e.g., if sin A = x, then sin⁻¹(x) = A.

Triangle Solution Requirements

Solve a right triangle by knowing two sides or one side and one acute angle.
Law of Sines applicable for any triangle, given two angles and any side (AAS or ASA).
Law of Cosines used when two sides and the included angle (SAS) or all three sides (SSS) are known.

Angles of Elevation and Depression

Angle of Elevation: measured from the ground up, upward angle from a horizontal line.
Angle of Depression: angle from horizontal line to a point below, observer's line of sight.

Geometric Mean Application

Used in scenarios with one large triangle containing two component triangles to find proportional parts.

Measurement Consistency

Ensure all measurements are in the same units for mathematical accuracy, such as converting km to m.

Triangle Ratios

When solving 45-45-90 triangles, the ratios are 1:1 for legs and √2 for hypotenuse.
For 30-60-90 triangles, the ratios are 1 (opposite 30°), 2 (hypotenuse), and √3 (opposite 60°).

Matching Pairs and Triangles

A matching pair consists of a side with its opposite angle having corresponding numerical measurements.
A hypotenuse is always present in right triangles, fundamental to utilizing the Pythagorean Theorem and related theorems.

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Description

This quiz covers key concepts in geometry, focusing on fractions, geometric means, and triangle properties. You'll explore the relationships between these concepts and apply theorems related to right triangles and geometric means. Prepare to test your understanding of the geometric mean and triangle properties!