Question

Seg AD is a tangent segment and line AC is a secant ICAB - 2.5 and BC - 75 Find AD.
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Answer 1

Answer:

(1) It is given that line AB is tangent to the circle at A.

∴ ∠CAB = 90º     (Tangent at any point of a circle is perpendicular to the radius throught the point of contact)

Thus, the measure of ∠CAB is 90º.

(2) Distance of point C from AB = 6 cm    (Radius of the circle)

(3) ∆ABC is a right triangle. 

CA = 6 cm and AB = 6 cm

Using Pythagoras theorem, we have

BC2=AB2+CA2⇒BC=62+62 ⇒BC=62  cm" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: 400; line-height: normal; font-size: 15px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(0, 0, 0); font-family: Arial, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;">BC2=AB2+CA2⇒BC=62+62−−−−−−√ ⇒BC=62–√  cmBC2=AB2+CA2⇒BC=62+62 ⇒BC=62  cm

Thus, d(B, C) = 62" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: 400; line-height: normal; font-size: 15px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(0, 0, 0); font-family: Arial, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;">62–√62 cm

(4) In right ∆ABC, AB = CA = 6 cm

∴ ∠ACB = ∠ABC      (Equal sides have equal angles opposite to them)

Also, ∠ACB + ∠ABC = 90º            (Using angle sum property of triangle)

∴ 2∠ABC = 90º

⇒ ∠ABC = 90°2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: 400; line-height: normal; font-size: 15px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(0, 0, 0); font-family: Arial, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;">90°290°2 = 45º

Thus, the measure of ∠ABC is 45º.