An inscribed angle in a circle is an angle whose vertex is on the top of the circle (on the circumference of the circle) and whose ends are chords in a circle.
![B - Angle inscribed in a circle](/_ipx/f_png,s_500x497/https://cdn.tutorela.com/images/B_-_Angle_inscribed_in_a_circle.width-500.png)
In which of the circles is the point marked in the circle and not on the circumference?
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 \( a \).
In which of the circles is the segment drawn the radius?
How many times longer is the radius of the red circle than the radius of the blue circle?
How many times longer is the radius of the red circle, which has a diameter of 24, than the radius of the blue circle, which has a diameter of 12?
In which of the circles is the point marked in the circle and not on the circumference?
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.
A circle has the following equation:
Point O is its center and is in the second quadrant ()
Use the completing the square method to find the center of the circle and its radius in terms of .
Let's recall that the equation of a circle with its center at and its radius is:
Now, let's now have a look at the equation for the given circle:
We will try rearrange this equation to match the circle equation, or in other words we will ensure that on the left side is the sum of two squared binomial expressions, one for x and one for y.
We will do this using the "completing the square" method:
Let's recall the short formula for squaring a binomial:
We'll deal separately with the part of the equation related to x in the equation (underlined):
We'll isolate these two terms from the equation and deal with them separately.
We'll present these terms in a form similar to the form of the first two terms in the shortcut formula (we'll choose the subtraction form of the binomial squared formula since the term in the first power we are dealing with is, which has a negative sign):
Notice that compared to the short formula (which is on the right side of the blue arrow in the previous calculation), we are actually making the comparison:
Therefore, if we want to get a squared binomial form from these two terms (underlined in the calculation), we will need to add the term, but we don't want to change the value of the expression, and therefore we will also subtract this term from the expression.
That is, we will add and subtract the term (or expression) we need to "complete" to the binomial squared form,
In the following calculation, the "trick" is highlighted (two lines under the term we added and subtracted from the expression),
Next, we'll put the expression in the squared binomial form the appropriate expression (highlighted with colors) and in the last stage we'll simplify the expression:
Let's summarize the steps we've taken so far for the expression with x.
We'll do this within the given equation:
We'll continue and do the same thing for the expressions with y in the resulting equation:
(Now we'll choose the addition form of the squared binomial formula since the term in the first power we are dealing with has a positive sign)
In the last step, we move the free numbers to the second side and combine like terms.
Now that the given circle equation is in the form of the general circle equation mentioned earlier, we can easily extract both the center of the given circle and its radius:
In the last step, we made sure to get the exact form of the general circle equation—that is, where only subtraction is performed within the squared expressions (emphasized with an arrow)
Therefore, we can conclude that the center of the circle is at: and extract the radius of the circle by solving a simple equation:
Remember that the radius of the circle, by its definition is the distance between any point on the diameter and the center of the circle. Since it is positive, we must disqualify one of the options we got for the radius.
To do this, we will use the remaining information we haven't used yet—which is that the center of the given circle O is in the second quadrant.
That is:
O(x_o,y_o)\leftrightarrow x_o<0,\hspace{4pt}y_o>0 (Or in words: the x-value of the circle's center is negative and the y-value of the circle's center is positive)
Therefore, it must be true that:
\begin{cases} x_o<0\rightarrow (x_o=4a)\rightarrow 4a<0\rightarrow\boxed{a<0}\\ y_o>0\rightarrow (y_o=-5a)\rightarrow -5a>0\rightarrow\boxed{a<0} \end{cases}
We concluded that a<0 and since the radius of the circle is positive we conclude that necessarily:
Let's summarize:
Therefore, the correct answer is answer d.
In which of the circles is the segment drawn the radius?
How many times longer is the radius of the red circle than the radius of the blue circle?
5
How many times longer is the radius of the red circle, which has a diameter of 24, than the radius of the blue circle, which has a diameter of 12?
2
How many times longer is the radius of the red circle than the radius of the blue circle?
Calculate the length of the arc marked in red given that the circumference is 6.
Calculate the length of the arc marked in red given that the circumference is 36.
Calculate the length of the arc marked in red given that the circumference is 12.
Calculate the length of the arc marked in red given that the circumference is 12.
How many times longer is the radius of the red circle than the radius of the blue circle?
Calculate the length of the arc marked in red given that the circumference is 6.
Calculate the length of the arc marked in red given that the circumference is 36.
2
Calculate the length of the arc marked in red given that the circumference is 12.
2
Calculate the length of the arc marked in red given that the circumference is 12.
8
Calculate the length of the arc marked in red given that the circumference is equal to 24.
Calculate the area of the section painted red given that the area of the circle is 12.
How many times longer is the radius of the red circle than the radius of the blue circle?
How many times longer is the radius of the red circle (14 cm) than the radius of the blue circle, which has a diameter of 7?
Calculate the length of the arc marked in red given that the circumference is 18.
Calculate the length of the arc marked in red given that the circumference is equal to 24.
Calculate the area of the section painted red given that the area of the circle is 12.
8
How many times longer is the radius of the red circle than the radius of the blue circle?
How many times longer is the radius of the red circle (14 cm) than the radius of the blue circle, which has a diameter of 7?
4
Calculate the length of the arc marked in red given that the circumference is 18.
13