Mathematics
In the following figure, PQ and PR are the tangent to the circle with centre O. Line segment AC touch the circle at point B.
(i) State the relation between tangent PQ and PR.
Also show that:
(ii) PQ = PA + AB
(iii) PR = PC + CB
(iv) PQ + PR = Perimeter of ΔPAC.
Answer
(i) PQ and PR are the tangent to the circle with centre O.
As we know that,
Through any point, outside a circle at most two tangent can be drawn and both of these tangent are always equal to each other.
Here, point P lies outside the circle and the two tangent PQ and PR are drawn to the circle.
Hence, PQ = PR.
(ii) When point A lies outside the circle and the two tangent AQ and AB are drawn to the circle. Then,
AQ = AB ………………..(1)
From figure,
⇒ PQ = PA + AQ
From equation (1),
⇒ PQ = PA + AB
Hence, proved that PQ = PA + AB.
(iii) When point C lies outside the circle and the two tangent CR and CB are drawn to the circle. Then
CR = CB ………………..(2)
From figure,
⇒ PR = PC + CR
From equation (2),
⇒ PR = PC + CB
Hence, proved that PR = PC + CB.
(iv) Perimeter of triangle PAC = PA + PC + AC
= PA + PC + (AB + CB)
= PA + PC + (AQ + CR)
= (PA + AQ) + (PC + CR)
= PQ + PR
Hence, proved that PQ + PR = Perimeter of Δ PAC.
Related Questions
In the following figure, O is the centre of the circle and AB is a diameter.
C and D are the points on the circumference of the circle such that ∠CAB = 35° and ∠ABD = 55°. Find the measures of angles CAD and CBD.
The following figure shows a circle with centre O and diameter PQ. Point R lies on the circumference of the circle such that PR = QR and PR = 4 cm.
Calculate the radius of the given circle and state its value correct to two decimal places.
From an external point P, the tangent PA is drawn to the circle with centre O.
(i) If OP = 20 cm and tangent PA = 16 cm find the diameter of the circle.
(ii) If diameter of the circle is 20 cm and tangent PA = 24 cm, find the length of OP.
In figure given below, O is the centre of the circle, use the given information to find the value of x:
