17. In the given figure, AB is a diameter of the circle with centre 'O', P is a point from where two tangents PT and PT' are drawn on the circle, a semicircle is drawn taking BC as diameter, A as centre and AP= PC. If AB = 12 cm, then the length of PD is
(1) 8 cm
(2) 12 cm
(3) 12 √2 cm
(4) 6√3 cm
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Step-by-step explanation:
Sure, please mark me as brainlist and follow me
To find the length of PD, we can use the property that tangents drawn from an external point to a circle are equal in length. Since PT and PT' are tangents to the circle, we know that PT = PT'.
Now, let's look at the semicircle. Since AP = PC, we can divide the semicircle into two equal parts. This means that angle APC is a right angle.
Since AB is the diameter of the circle, angle APB is also a right angle.
Now, we have two right triangles: APB and APC.
Using the Pythagorean theorem in triangle APB, we have:
AB^2 = AP^2 + PB^2
Substituting the values, we get:
12^2 = AP^2 + PB^2
Simplifying, we have:
144 = AP^2 + PB^2
Since AP = PC, we can substitute PC for AP:
144 = PC^2 + PB^2
Now, let's look at triangle APC. Since angle APC is a right angle, we can use the Pythagorean theorem again:
PC^2 = PD^2 + CD^2
Since AP = PC, we can substitute AP for PC:
AP^2 = PD^2 + CD^2
But we know that AP = PC, so we can substitute PC for AP:
PC^2 = PD^2 + CD^2
Now we have two equations:
144 = PC^2 + PB^2
PC^2 = PD^2 + CD^2
Since AP = PC, we can substitute AP for PC in the first equation:
144 = AP^2 + PB^2
Substituting the value of AP, we get:
144 = 12^2 + PB^2
Simplifying, we have:
144 = 144 + PB^2
Subtracting 144 from both sides, we get:
0 = PB^2
This means that PB = 0.
Since PB = 0, PD = PC.
Therefore, the length of PD is equal to the length of PC.
Since AP = PC, and AP = 12 cm, the length of PD is also 12 cm.
So, the correct answer is (2) 12 cm.
Verified answer
Answer:
6√3 cm
Step-by-step explanation:
ATQ,
AB=12 CM
=> AO=AB=6CM
ALSO
BC IS DIAMETER SO AND 'A' IS CENTRE
=> AC = BC = 12 CM
ALSO AP=AC
=> AP=AC=6 CM
AP=AC=AO=AB=6CM
Hope you have understood this (because this is too simple)
And now comes the good part...
Let x=6cm
so the figure would like this (refer to first image)
after dividing all the x by x the figure would look like this( refer to second image)
as it's visible
PB= PA+AO+OB
PB = 1+1+1 = 3
and CP = 1
Now we have studied a proof in the ninth class that in such a given figure,
PD=√PB (refer to 3rd image)
Soooo
PD = √3
But now we will have to multiply RHS by x because we divided by x earlier
PD= √3 × x
But x=6 cm
=> PD =6√3 cm
BEAUTIFUL RIGHT
And in case you are wondering, the tangents might have been given to distract your mind :)
(please mark as brainliest)