Question

In the semi-circle with centre O, PO ⊥ diameter AB. ∠ACB = 90°, AC = 24 cm and CQ = 7 cm. Find the length of BQ, AB and PQ.

Answer 1

Given:

Semi-circle with centre O

• PO ⊥ AB
• ∠ACB = 90°
• AC = 24 cm 
• CQ = 7 cm

We need to find:

1. Length of BQ
2. Length of AB
3. Length of PQ

Step 1: Use triangle ΔACQ to find AQ

Using Pythagoras Theorem:

AQ² = AC² – CQ²

24² – 7²

= 576 – 49

= 527

⇒ `AQ = sqrt(527) ≈ 22.96`

Step 2: Use symmetry to find BQ

Since PQ ⊥ AN and goes through center O, it bisects the diameter:

⇒ AQ = BQ

= `sqrt(527)` ≈ \[\boxed{\text{25 cm}}\]

Step 3: Find AB

AB = AQ + BQ

= `sqrt(527) + sqrt(527)`

= `2sqrt(527)` ≈ \[\boxed{\text{40 cm}}\]

Step 4: Find PQ

Since O is midpoint of diameter:

AO = BO = `(AB)/2` = 20 cm

AQ = 25 cm

So, OQ = AQ – AO = 25 – 20 = 5 cm

Since PQ ⊥ AB and passes through center:

PQ = OQ = \[\boxed{\text{5 cm}}\]