AC is the diameter of the semicircle and BD is perpendicular to AC. If AB = 8 cm and BC = 2 cm, find the length of BD.


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Answer:

4 cm

Step by Step Explanation:
  1. The following figure shows the semicircle with the diameter AC,

    According to the question, AB = 8 cm, and BC = 2 cm.
    The diameter(AC) of the semicircle = AB + BC
    = 8 + 2
    = 10 cm
  2. Let's draw a line OD from the center of the semicircle, as shown in the following figure,

    The line OD, OA and OC are the radii of the semicircle.
    Therefore, OD = OA = OC =  
    AC
    2
     
    =  
    10
    2
     
    = 5 cm
  3. Now, OB = OC - BC
    = 5 - 2
    = 3 cm
  4. In ΔOBD, ∠OBD = 90°
    BD2 = OD2 - OB2 [By the Pythagorean theorem]
    ⇒ BD2 = 52 - 32
    ⇒ BD2 = 16
    ⇒ BD2 = 42
    ⇒ BD = 4 cm
  5. Hence, the length of the BD is 4 cm.

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