(2) In the given Fig 10.86 '0 ' is the centre of
circle & PQR = 105°. Find the
x.
Answers
Answer:Question 1:
If PT is a tangent at T to a circle whose centre is O and OP = 17 cm, OT = 8 cm, Find the length of the tangent segment PT.
ANSWER:
Let us put the given data in the form of a diagram.
We have to find TP. From the properties of tangents we know that a tangent will always be at right angles to the radius of the circle at the point of contact. Therefore is a right angle and triangle OTP is a right triangle.
We can find the length of TP using Pythagoras theorem. We have,
Therefore, the length of TP is 15 cm.
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Question 2:
Find the length of a tangent drawn to a circle with radius 5 cm, from a point 13 cm from the centre of the circle.
ANSWER:
Let us first put the given data in the form of a diagram.
We have to find TP. From the properties of tangents we know that a tangent will always be at right angles to the radius of the circle at the point of contact. Therefore is a right angle and triangle OTP is a right triangle.
We can find the length of TP using Pythagoras theorem. We have,
Therefore, the length of TP is 12 cm.
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Question 3:
A point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 10 cm. Find the radius of the circle.
ANSWER:
Let us put the given data in the form of a diagram.
We have to find OT. From the properties of tangents we know that a tangent will always be at right angles to the radius of the circle at the point of contact. Therefore is a right angle and triangle OTP is a right triangle.
We can find the length of TP using Pythagoras theorem. We have,
Therefore, the radius of the circle is 24 cm.
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Question 4:
If from any point on the common chord of two interesting circle, tangents be drawn to the circles, prove that they are equal.
ANSWER:
Let the two circles intersect at points X and Y. XY is the common chord.
Suppose A is a point on the common chord and AM and AN be the tangents drawn from A to the circle.
We need to show that AM = AN.
In order to prove the above relation, following property will be used.
“Let PT be a tangent to the circle from an external point P and a secant to the circle through P intersects the circle at points A and B, then PT 2 = PA × PB”.
Now, AM is the tangent and AXY is a secant.
∴ AM2 = AX × AY ...(1)
AN is the tangent and AXY is a secant.
∴ AN2 = AX × AY ...(2)
From (1) and (2), we have
AM2 = AN2
∴ AM = AN
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Question 5:
If the sides of a quadrilateral touch a circle. prove that the sum of a pair of opposite sides is equal to the sum of the other pair.
ANSWER:
Let us first put the given data in the form of a diagram.
We have been asked to prove that the sum of the pair of opposite sides of the quadrilateral is equal to the sum of the other pair.
Therefore, we shall first consider,
AB + DC
But by looking at the figure we have,
AB + DC = AF + FB + DH + HC …… (1)
From the property of tangents we know that the length of two tangents drawn to a circle from a common external point will be equal. Therefore we have the following,
AF = AE
FB = BG
DH = ED
HC = CG
Replacing for all the above in equation (1), we have
AB + DC = AE + BG + ED + CG
AB + DC = (AE + ED) + (BG +CG)
AB + DC = AD + BC
Thus we have proved that the sum of the pair of opposite sides of the quadrilateral is equal to the sum of the other pair.
Step-by-step explanation: