In a ∆ ABC, AB = AC. A circle through B touches AC at D and intersec

1.	In a ∆ ABC, AB = AC. A circle through B touches AC at D and intersects AB at P. If D is the mid-point of AC, which one of the following is correct? (a) AB = 2AP (b) AB = 3AP (c) AB = 4AP (d) 2AB = 5AP
Answer» Answer : (c) AB = 4AP Using the tangent-secant theorem, we have AB × AP = AD² = \(\big(\frac{AC}{2}\big)^2\) ( ∵ AD = DC) ⇒ AB × AP = \(\frac{1}{4}\) AC² = \(\frac{1}{4}\) AB² ( ∵ AB = AC) ⇒ AB = 4 AP

In a ∆ ABC, AB = AC. A circle through B touches AC at D and intersects AB at P. If D is the mid-point of AC, which one of the following is correct? (a) AB = 2AP (b) AB = 3AP (c) AB = 4AP (d) 2AB = 5AP

Answer»

Answer : (c) AB = 4AP

Using the tangent-secant theorem, we have

AB × AP = AD² = \(\big(\frac{AC}{2}\big)^2\) ( ∵ AD = DC)

⇒ AB × AP = \(\frac{1}{4}\) AC² = \(\frac{1}{4}\) AB² ( ∵ AB = AC)

⇒ AB = 4 AP

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In a ∆ ABC, AB = AC. A circle through B touches AC at D and intersects AB at P. If D is the mid-point of AC, which one of the following is correct? (a) AB = 2AP (b) AB = 3AP (c) AB = 4AP (d) 2AB = 5AP

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