# Please help Explain how I figure this out.

A circle has diameter 70cm.

A chord in the circle is 50cm long.

How far is the chord from the center of the circle?

I have been stuck on this for days. I appreciate any help. Thank you!!

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## To figure out how far the chord is from the center of the circle, you can use the theorem that states that the perpendicular drawn from the center of a circle to a chord bisects the chord.

Here's how you can solve the problem step-by-step:

Step 1: Start by drawing a circle with a diameter of 70 cm. This circle represents the given scenario.

Step 2: Draw a chord in the circle that has a length of 50 cm. Label the endpoints of the chord as A and B.

Step 3: Locate the midpoint of the chord by measuring half of its length. In this case, the midpoint of the chord is at a distance of 25 cm from point A. Label this midpoint as M.

Step 4: Draw a line from the center of the circle to the midpoint of the chord. Label the point where this line intersects the chord as C.

Step 5: Now, you have a right-angled triangle formed by the line from the center of the circle to the midpoint of the chord, the chord itself, and the radius of the circle.

Step 6: In this triangle, the chord serves as the hypotenuse, and the line from the center of the circle to the midpoint of the chord is the height from the center. The radius is the other leg of the right-angled triangle.

Step 7: You know that the length of the chord is 50 cm, and the diameter of the circle is 70 cm. So, the radius of the circle is half the diameter, which is 35 cm.

Step 8: Since the line from the center of the circle to the midpoint of the chord is perpendicular to the chord, it acts as the height of the triangle. It divides the chord into two equal parts, making each part 25 cm long.

Step 9: Now, you have a right-angled triangle with one leg measuring 25 cm (half of the chord), and the other leg measuring 35 cm (the radius of the circle).

Step 10: Using the Pythagorean theorem (a^2 + b^2 = c^2), you can find the distance from the center of the circle to the chord. In this case, it is:

25^2 + 35^2 = c^2

625 + 1225 = c^2

1850 = c^2

Step 11: Take the square root of both sides of the equation to solve for c:

c = √1850

Calculating the square root of 1850 gives you approximately 43.01 cm.

So, the distance from the chord to the center of the circle is approximately 43.01 cm.

## To find the distance between the chord and the center of the circle, we can use the following steps:

1. Draw a diagram: Start by drawing a diagram to better visualize the problem. Draw a circle with a diameter of 70cm. Then draw a chord inside the circle that measures 50cm.

2. Identify the relevant points: Label the center of the circle as point O, and the midpoint of the chord as point M. The task is to find the distance between point M and point O.

3. Recall the properties of a perpendicular bisector: In a circle, the perpendicular bisector of a chord passes through the center of the circle. This means that if we can find the midpoint of the chord, we can extend a line from the midpoint to the center, forming a right angle.

4. Find the midpoint of the chord: Since the length of the chord is given as 50cm, and we know it is the diameter of the circle, it means the midpoint of the chord is also the center of the circle. Therefore, point M is the center of the circle, which we labeled as point O.

5. Measure the distance: Now that we know point M is the same as point O, the distance between the chord and the center of the circle is zero. In other words, the chord lies directly on the center of the circle.

Therefore, the distance from the chord to the center of the circle is 0cm.

## draw a diameter through the center of the chord.

draw a radius to the end of the chord.

now, the distance d from the chord to the center can be found using

d^2 + 25^2 = 35^2