A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the ∆ABC
Solution:
Given, A is a point at a distance of 13 cm from the centre O of a circle of radius 5 cm.
AP and AQ are the tangents to the circle at P and Q.
A tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C.
We have to find the perimeter of the triangle ABC.
From the figure,
Radius of the circle = OP = OQ = 5 cm
Given, OA = 13 cm
We know that the tangents drawn to a circle through an external point are equal.
Given, AP and AQ are the tangents to the circle from an external point A.
So, AP = AQ ------------------- (1)
We know that the radius of the circle is perpendicular to the tangent at the point of contact.
So, OP ⟂ PA and OQ ⟂ AQ
Also, ∠OPA = ∠OQA = 90°
Considering triangle OPA,
OPA is a right triangle with P at right angle.
OA² = OP² + PA²
(13)² = (5)² + PA²
169 = 25 + PA²
PA² = 169 - 25
PA² = 144
Taking square root,
PA = 12 cm
Perimeter of a triangle is the sum of all three sides of the triangle.
Perimeter of triangle ABC = AB + BC + AC
From the figure,
BC = BR + CR
Perimeter of triangle ABC = AB + BR + CR + AC
We know that the tangents drawn to a circle through an external point are equal.
Tangents from external point B are BP and BR
So, BP = BR
Tangents from external point C are CQ and CR
So, CQ = CR
Perimeter of triangle ABC = AB + BR + CR + AC
= AB + BP + CQ + AC
We know AP = AB + BP
So, Perimeter of triangle ABC = AP + CQ + AC
From the figure,
AQ = AC + CQ
Now, Perimeter of triangle ABC = AP + AQ
From (1), AP + AP = 2 AP
Perimeter of triangle ABC = 2AP
= 2(12)
= 24 cm
Therefore, the perimeter of triangle ABC is 24 cm.
✦ Try This: Let ABCD be a cyclic quadrilateral and let P and Q be points on the sides AB and AD respectively, such that AP = CD and AQ = BC. Let M be the point of intersection of AC and PQ. Then, show that M is the midpoint of PQ.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10
NCERT Exemplar Class 10 Maths Exercise 9.4 Problem 14
A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the ∆ABC
Summary:
A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, the perimeter of the ∆ABC is 24 cm
☛ Related Questions:
- In Fig. 9.20. O is the centre of a circle of radius 5 cm, T is a point such that OT = 13 cm and OT i . . . .
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