A circle R has a radius of length 5 sqrt 3 and has a chord XY that is 5 units from R. Find the length of XY.$
Draw a diagram. If half the chord has length x, you have a right triangle.
5^2+x^2 = (5√3)^2
x^2+25=75
x^2=50
x = 5√2
So, XY = 10√2
Thanks Steve. You're the best!
To find the length of chord XY, we can use the properties of a circle.
First, let's draw a diagram to visualize the problem.
```
R
/ \
/ \
/ \
/ \
/ \
/ \
X-----Y
```
Given that the radius of circle R is 5√3 and that XY is 5 units away from R, we can conclude that XY is a chord of the circle.
Now, let's consider the right triangle formed by the radius, the perpendicular from center R to XY, and the chord XY.
```
R
/ \
/ \
/ | \
/ | \
/ | \
/_____|_____\
X-----P-----Y
```
In this right triangle, RP is the perpendicular from center R to XY.
We know that RP is 5 units (5 units from R to XY). The radius of circle R is also given as 5√3.
Using the Pythagorean theorem, we can find the length of XY:
```
(RP)^2 + (XY/2)^2 = (R)^2
(5)^2 + (XY/2)^2 = (5√3)^2
25 + (XY/2)^2 = 75
(XY/2)^2 = 75 - 25
(XY/2)^2 = 50
XY/2 = √50
XY = 2√50
XY = 2√(25 * 2)
XY = 2√25√2
XY = 2 * 5√2
XY = 10√2 units
```
Therefore, the length of XY is 10√2 units.
To find the length of chord XY, we can use the Pythagorean theorem.
Let's break down the problem step by step:
1. Draw a circle R with a radius of length 5√3.
- We know that the radius is the distance from the center of the circle to any point on the circle.
2. Draw a chord XY that is 5 units from the center of the circle, R.
- The chord is a line segment that connects two points on the circle.
- In this case, it is 5 units away from the center.
Now, we need to find the length of the chord XY. To do this, we can use the Pythagorean theorem.
3. Label the point where the chord XY intersects the radius as point A.
- This point A divides the chord XY into two equal segments.
4. Draw a line segment from the center of the circle to the midpoint of chord XY.
- This line segment is perpendicular to the chord and it bisects the chord.
5. Label the midpoint as point B. The line segment from the center to the midpoint is also the radius of the circle.
- We can use the Pythagorean theorem to find the length of segment AB.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
6. In triangle RAB, we have the following:
- The hypotenuse RA, which is the radius of the circle, has a length of 5√3.
- The side AB, which is the distance from the center to the midpoint of the chord, is half the length of the chord XY, which we want to find.
- The side RB is the perpendicular distance from the center to the chord XY, which is given as 5 units.
Applying the Pythagorean theorem, we have:
(AB)^2 + (RB)^2 = (RA)^2
Let's substitute the given values:
(AB)^2 + (5)^2 = (5√3)^2
Simplifying the equation, we get:
(AB)^2 + 25 = 75
Subtracting 25 from both sides, we have:
(AB)^2 = 50
Finally, taking the square root of both sides to solve for AB, we get:
AB = √50
Simplifying √50, we have:
AB = √(25 * 2) = 5√2
So the length of the chord XY is 2 * AB, which is:
XY = 2 * (5√2) = 10√2
Therefore, the length of the chord XY is 10√2 units.