Math Problem Statement

AD is the diameter of a circle of radius 17cm and AB is of length 30 cm. Find the distance of AB from the centre.

Solution

To find the distance of line segment ABAB from the center of the circle, we can use the following steps:

  1. Identify key components:

    • The radius of the circle r=17r = 17 cm.
    • The diameter AD=2×r=34AD = 2 \times r = 34 cm.
    • The length of segment AB=30AB = 30 cm.
  2. Use the right triangle relationship:

    • Let OO be the center of the circle.
    • The distance from OO to ABAB is the perpendicular distance we want to find, denoted as dd.
    • The midpoint MM of ABAB is where the perpendicular from OO meets ABAB.
  3. Apply the Pythagorean theorem:

    • The distance OAOA from the center OO to point AA is equal to the radius, which is 1717 cm.
    • The length AMAM (half of ABAB) is 302=15\frac{30}{2} = 15 cm.

Using the Pythagorean theorem in triangle OAMOAM: OA2=OM2+AM2OA^2 = OM^2 + AM^2 172=d2+15217^2 = d^2 + 15^2 289=d2+225289 = d^2 + 225 d2=289225=64d^2 = 289 - 225 = 64 d=64=8 cmd = \sqrt{64} = 8 \text{ cm}

Thus, the distance of ABAB from the center of the circle is 8 cm.

Do you want details or have any questions?

Here are some related questions you might find interesting:

  1. What would happen to the distance if the length of ABAB were greater than the diameter?
  2. How do you find the area of the circle given the radius?
  3. Can you explain how to find the angle subtended by chord ABAB at the center?
  4. What are other methods to find the distance from a point to a line?
  5. How would the situation change if ABAB were a secant line instead?

Tip: Always remember to check whether a line segment lies entirely within the circle or intersects it to apply the right geometric principles.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Properties
Pythagorean Theorem

Formulas

Diameter = 2 * Radius
Pythagorean Theorem: OA^2 = OM^2 + AM^2

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 7-9