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.

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

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

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 $$ a $$.

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

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

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

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

Perpendicular to Chord from Circle Center Practice Problems

Master perpendicular properties from circle center to chord with step-by-step practice problems. Learn chord bisection, angle bisection, and arc properties.

📚Practice Perpendicular from Circle Center to Chord Properties

Apply the chord bisection property when perpendicular passes through center
Calculate equal segments when perpendicular bisects a chord
Find equal central angles created by perpendicular from center
Determine equal arc lengths on opposite sides of perpendicular
Solve problems using all three perpendicular properties together
Identify when a line from center is perpendicular to chord

Understanding Perpendicular to a chord from the center of a circle

Complete explanation with examples

The perpendicular to the chord comes out of the center of the circle, intersecting the chord, the central angle in front of the chord and the arc in front of the chord.
Moreover, if there is a section that comes out from the center of the circle and crosses the chord, it will also be perpendicular to the chord.

We are here to present the properties of the perpendicular from the center of the circle to the chord.
First, we will remember that the perpendicular is a line that forms a $90°$ degree angle.
Let's see it in the illustration:

A1 - Perpendicular to the chord from the center of the circle

In front of us, there is a circle.
We will mark the center of the circle with a letter $A$
Our chord will be blue and will be called $BC$ .
The vertical, which comes out from the center of the circle and will be perpendicular to the chord $BC$ .
We will mark it in red and call it $AD$ .

Detailed explanation

Practice Perpendicular to a chord from 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 Perpendicular to a chord from the center of a circle

Step-by-step solutions included

Exercise #1

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

Exercise #2

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 #3

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 #4

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 #5

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:

Frequently Asked Questions

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

What are the three main properties of a perpendicular from the center of a circle to a chord?

The three properties are: 1) It bisects the chord into two equal segments, 2) It bisects the central angle in front of the chord, and 3) It bisects the arc that corresponds to the chord. These properties work together and can be used to solve various circle geometry problems.

How do you prove that a perpendicular from the center bisects a chord?

When a line from the center of a circle is perpendicular to a chord, it creates two congruent right triangles. Since the radii are equal and the perpendicular creates equal angles, the triangles are congruent by RHS, making the chord segments equal.

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

The closer a chord is to the center of the circle, the longer the chord becomes. When the perpendicular distance from center to chord decreases, the chord length increases, with the diameter being the longest possible chord.

Can you use perpendicular properties to find missing chord segments?

Yes, if you know one segment of a chord that's bisected by a perpendicular from the center, the other segment is equal. For example, if BD = 5 cm and the perpendicular bisects chord BC, then DC = 5 cm as well.

How does the perpendicular from center affect central angles?

The perpendicular from the center to a chord bisects the central angle, creating two equal angles. If the original central angle is 60°, the perpendicular divides it into two 30° angles.

What happens to arc lengths when perpendicular bisects the chord?

The perpendicular from center to chord also bisects the corresponding arc, creating two equal arc segments. This means the arc lengths on either side of the perpendicular are identical.

When is a line from the circle center automatically perpendicular to a chord?

If a line from the center of a circle bisects a chord, then it is automatically perpendicular to that chord. This is the converse property - any of the three main properties implies the others.

How do you solve problems involving multiple chords and perpendiculars?

For multiple chords: 1) Identify which perpendiculars come from the center, 2) Apply the three properties to each relevant chord, 3) Use the equal segments, angles, and arcs to set up equations, 4) Solve systematically using the given information.

Continue Your Math Journey

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

Topics Learned in Later Sections

Circle Distance from a chord to the center of a circle Chords of a Circle Central Angle in a Circle Arcs in 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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