From the given figure, prove that : AP + BQ + CR = BP + CQ + AR. Also, show that : AP + BQ + CR = x Perimeter of triangle ABC.

Question

KnowledgeBoat · Accepted Answer

We know that, If two tangents are drawn to a circle from an exterior point, the tangents are equal in length. From point B, BQ and BP are the tangents to the circle BQ = BP ........ (1) From point A, AP and AR are the tangents to the circle AP = AR ........ (2) From point C, CR and CQ are the tangents to the circle CR = CQ ........ (3) Adding (1), (2) and (3) we get, AP + BQ + CR = BP + CQ + AR ......... (4) Hence, proved that AP + BQ + CR = BP + CQ + AR. Now, adding AP + BQ + CR to both sides in (4), we get 2(AP + BQ + CR) = AP + BP + CQ + BQ + AR + CR ..........(5) From figure, AP + BP = AB, BQ + CQ = BC and AR + CR = AC. Substituting above value in equation (5), we get : ⇒ 2(AP + BQ + CR) = AB + BC + CA ⇒ AP + BQ + CR = (AB + BC + CA). ⇒ AP + BQ + CR = (Perimeter of △ABC). [∵ Perimeter = AB + BC + CA] Hence, proved that AP + BQ + CR = (Perimeter of △ABC).

Mathematics

From the given figure, prove that : AP + BQ + CR = BP + CQ + AR. Also, show that : AP + BQ + CR = x Perimeter of triangle ABC.

Circles

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If the sides of a quadrilateral ABCD touch a circle, prove that AB + CD = BC + AD.

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Mathematics

From the given figure, prove that : AP + BQ + CR = BP + CQ + AR. Also, show that : AP + BQ + CR = x Perimeter of triangle ABC.

Circles

Related Questions

In the given figure, O is the centre of the circle and AB is a tangent to the circle at B. If AB = 15 cm and AC = 7.5 cm, calculate the radius of the circle.

If the sides of a quadrilateral ABCD touch a circle, prove that AB + CD = BC + AD.

In the given figure, if AB = AC then prove that BQ = CQ.

Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if :(i) they touch each other externally,(ii) they touch each other internally.

Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centers if :
(i) they touch each other externally,
(ii) they touch each other internally.