Question

In ABC, AB = AC. A circle drawn through B touches AC at D and intersect AB at P. If D is the mid point of AC and
AP= 2.5 cm, then AB is equal to:
ans 10 cm​

Answer 1

Question:

In ∆ABC, AB = AC. A circle drawn through B touches AC at D and intersect AB at P. If D is the mid point of AC and AP= 2.5 cm, then AB is equal to?

Answer

AB = 10 cm

Explanation

By Tangent-Secant Theorem

“When a tangent and secant drawn from an external point of circle. Square of length of tangent is equal to product of secant of segment and and length of segment.”

⇒ AD² = AB.AP

Given that, D is a mid-point of AC

→ AD = AC/2

Substitute value of AD in AD²

→ (AC/2)² = AB.AP

→ (AC)²/4 = AB.AP

Alsog given that, AB = AC

→ (AB)²/4 = AB.AP

→ (AB)²/AB = 4AP

→ AB = 4AP

We have given the value of AP = 2.5 cm. Substitute the value of AP to find the value of AB.

→ AB = 4(2.5) cm

→ AB = 10 cm

Answer 2

||✪✪ QUESTION ✪✪||

In ABC, AB = AC. A circle drawn through B touches AC at D and intersect AB at P. If D is the mid point of AC and

AP= 2.5 cm, then AB is equal to ?

|| ✰✰ ANSWER ✰✰ ||

❁❁ Refer To Image First .. ❁❁

Given That :-

→ AB = AC (∆ABC is a isosceles ∆).

→ Circle Through B , cuts AB at P.

→ And AC at D

→ AD = DC . ( D is Mid - Point of AC)

Theorem Used :- The product of the length of the secant segment and its external part equals the square of the length of the tangent segment.

☛ AD² = AP * AB (From Image).

________

Using This Now :-

➪ AD² = AP * AB

Putting AD = (AC/2) Now,

➪ (AC/2)² = AP * AB

➪ AC²/4 = AP * AB

Now, As ∆ABC is a isosceles ∆ , with AB = AC, putting AB = AC in LHS,

➪ AB²/4 = AP * AB

➪ AB² = 4 * AP * AB

Dividing both sides by AB now,

➪ AB = 4 * AP

Putting value of AP = 2.5cm Now,

➪ AB = 4 * 2.5

➪ AB = 10cm.

ஃ Value of AB is 10cm .