A circle has the following equation: $ x^2-8ax+y^2+10ay=-5a^2 $ Point O is its center and is in the second quadrant ($$ a\neq0 $$) Use the completing the square method to find the center of the circle and its radius in terms of $$ a $$.

$$ O(4a,-5a),\hspace{4pt}R=-6a $$

$$ O(4a,5a),\hspace{4pt}R=6a $$

$$ O(4a,10a),\hspace{4pt}R=6a $$

$$ O(4a,-5a),\hspace{4pt}R=6a $$

A circle has the equation: $$ x^2-6x+y^2-4y=7 $$ Use the completing the square method to find the center of the circle and its radius.

$$ (3,2),\hspace{6pt} R=\sqrt{20} $$

$$ (3,-2),\hspace{6pt} R=\sqrt{20} $$

Circle Distance from Chord to Center Practice Problems

Master distance from chord to center theorems with step-by-step practice problems. Learn equal chords theorem and chord length relationships.

📚Practice Problems: Distance from Chord to Center

Apply the equal chords equidistant theorem to solve geometry problems
Calculate perpendicular distances from circle center to various chords
Use chord length and distance relationships to find unknown measurements
Compare chord lengths using distance from center theorem
Solve multi-step problems involving chord distances and circle properties
Apply reverse theorems to determine chord equality from equal distances

Understanding Distance from a chord to the center of a circle

Complete explanation with examples

The distance from the chord to the center of the circle is defined as the length of the perpendicular from the center of the circle to the chord.
Theorems on the distance from the center of the circle:

Chords that are equal to each other are equidistant from the center of the circle.
If in a circle, the distance of a chord from the center of the circle is less than the distance of another chord from the center of the circle, we can determine that the chord with the lesser distance is longer than the other chord.

A1 - The distance from the chord to the center of the circle

All theorems can also exist in reverse.

Detailed explanation

Practice Distance from a chord to the center of a circle

Test your knowledge with 6 quizzes

Calculate the length of the arc marked in red given that the circumference is 12.

Examples with solutions for Distance from a chord to the center of a circle

Step-by-step solutions included

Exercise #1

In which of the circles is the point marked inside of the circle and not on the circumference?

Step-by-Step Solution

Let's remember that the circular line draws the shape of the circle, and the inner part is called a disk.

Therefore, in diagram B, the point is located in the inner part, meaning inside the disk.

Answer:

Video Solution

Exercise #2

Identify which diagram shows the radius of a circle:

Step-by-Step Solution

Remember that a radius is a line segment connecting the center of a circle to any point on the circle itself.

In drawing C we can see that the line coming from the center of the circle indeed connects to a point on the circle itself, while in the other drawings the lines don't touch any point on the circle.

Therefore, C is the correct drawing.

Answer:

Exercise #3

Identify which diagram shows the radius of a circle:

Step-by-Step Solution

Remember that a radius is a line segment connecting the center of the circle to a point that lies on the circle itself.

In drawing A, the line doesn't touch any point on the circle itself.

In drawing B, the line doesn't pass through the center of the circle.

We can see that in drawing C, the line that extends from the center of the circle is indeed connected to a point on the circle itself.

Answer:

Video Solution

Exercise #4

Where does a point need to be so that its distance from the center of the circle is the shortest?

Step-by-Step Solution

Let's remember that the circle is actually the inner part of the circumference, meaning the enclosed area within the frame of the circumference.

Therefore, a point whose distance is less than the radius from the center of the circle will necessarily be inside the circle.

Answer:

Inside

Exercise #5

A point whose distance from the center of the circle is _______ than the radius, is outside the circle.

Step-by-Step Solution

Let's remember that the circle is actually the inner part of the circumference, meaning the enclosed area within the frame of the circumference.

Therefore, a point whose distance is greater than the center of the circle will necessarily be outside the circle.

Answer:

greater

Frequently Asked Questions

Everything you need to know about Distance from a chord to the center of a circle

How do you find the distance from a chord to the center of a circle?

The distance from a chord to the center of a circle is the length of the perpendicular line drawn from the center of the circle to the chord. This perpendicular line creates a right angle with the chord and represents the shortest distance between the center and the chord.

What is the equal chords theorem for distance from center?

The equal chords theorem states that chords of equal length are equidistant from the center of the circle. This means if two chords have the same length, their perpendicular distances from the center will also be equal. The theorem also works in reverse: if two chords are equidistant from the center, they must be equal in length.

How does chord length relate to distance from center?

There is an inverse relationship between chord length and distance from center. The closer a chord is to the center (shorter distance), the longer the chord will be. Conversely, the farther a chord is from the center (longer distance), the shorter the chord will be.

What are the main theorems about chord distances in circles?

The two main theorems are: 1) Equal chords are equidistant from the center, and 2) If one chord has a shorter distance from the center than another chord, then the first chord is longer than the second chord. Both theorems work in reverse as well.

Why is the perpendicular distance used to measure chord to center distance?

The perpendicular distance is used because it represents the shortest possible distance between a point (the center) and a line (the chord). Any other line from the center to the chord would be longer than the perpendicular, making it the most accurate and standardized measurement.

Can you use chord distance theorems to solve for unknown chord lengths?

Yes, you can use chord distance theorems to find unknown chord lengths. If you know the distances of chords from the center, you can compare their lengths using the inverse relationship theorem. If distances are equal, the chords are equal; if one distance is shorter, that chord is longer.

What is the difference between chord length and chord distance from center?

Chord length is the actual measurement of the chord segment itself, while chord distance from center is the perpendicular distance from the circle's center to that chord. These are two different measurements that are related through the chord distance theorems.

How do you remember the chord distance and length relationship?

Remember this simple rule: 'shorter distance - longer chord, longer distance - shorter chord.' This inverse relationship means that as a chord gets closer to the center, it becomes longer, and as it moves farther from the center, it becomes shorter.

Continue Your Math Journey

Master Distance from a chord to the center of a circle first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

Circle Chords of a Circle Central Angle in a Circle Arcs in a Circle Perpendicular to a chord from the center of a circle Inscribed angle in a circle Tangent to a circle Circuit Components Elements of the circumference Area of a circle The Circumference of a Circle How is the radius calculated using its circumference?

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