in the given figure, line AB is tangent to both the circles touching at A and B. OA = 29, BP = 18, OP = 61 then find AB
Answers
If OA = 29, BP = 18, OP = 61 then the length of tangent AB is 60 units.
Step-by-step explanation:
It is given that
AB is tangent to both the circles at A and B.
OA = 29 units
BP = 18 units
OP = 61 units
Step 1:
Referring to the figure attached below, we will draw a perpendicular line from point P to the radius OA of the bigger circle at M, i.e.,
∠AMP = 90° …. (i)
We know that the tangent to a circle is perpendicular to the radius at the point of tangency.
∴ ∠OAB = 90° and ∠PBA = 90° …. (ii)
Consider quadrilateral AMPB, applying angle sum property, we get
∠OAB + ∠PBA + ∠AMP + ∠BPM = 360°
⇒ 90° + 90° + 90° + ∠BPM = 360°
⇒ ∠BPM = 360° - 270°
⇒ ∠BPM = 90° …… (iii)
Step 2:
From (i), (ii) & (iii), we get
Quadrilateral AMPB is a rectangle
∴ AM = BP = 18 units …… [opposite sides of a rectangle are equal in length]
∴ OM = OA – AM = 29 – 18 = 11 units ….. (iv)
Now, in right-angled ∆ PMO, applying Pythagoras theorem,
OP² = OM² + PM²
⇒ 61² = 11² + PM² ….. [OP = 61 (given) and OM = 11 (from (iv))]
⇒ PM² = 3721 – 121
⇒ PM = √3600
⇒ PM = 60 units
∴ PM = AB = 60 ← opposite sides of rectangle AMPB are equal in length
Thus, the length of AB is 60 units.
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Step-by-step explanation:
in Above photo I given all explanations
