# a circle touches all the four sides of a quadrilateral abcd

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## In figure, a circle touches all the four sides of a quadrilateral ABCD , whose sides AB = 6 cm , BC = 7 cm , and CD = 4 cm . Length of AD is

Click here👆to get an answer to your question ✍️ In figure, a circle touches all the four sides of a quadrilateral ABCD , whose sides AB = 6 cm , BC = 7 cm , and CD = 4 cm . Length of AD is

Question

## In figure, a circle touches all the four sides of a quadrilateral ABCD, whose sides AB=6cm, BC=7cm, and CD=4cm. Length of AD is

**A**

## 10 cm

**B**

## 7 cm

**C**

## 3 cm

**D**

## 4 cm

Medium Open in App Solution Verified by Toppr

Correct option is C)

**Given :**A circle inscribed in a quadrilateral ABCD,AB=6cm,BC=7cm,CD=4cm.

**To find :**AD

Proof : Let the circle touch the sides AB,BC,CD,DA, at P,Q,R and S, respectively.

AP=AS BP=BQ DR=DS

CR=CQ {Lengths of two tangents drawn from an external point of circle, are equal}

Adding all these, we get

(AP+BP)+(CR+RD)=(BQ+QC)+(DS+SA)

AB+CD=BC+DA ⇒6+4=7+AD ⇒AD=10−7=3cm.

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## A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA

A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA

Byju's Answer Standard X Mathematics

Construction of Incircle

A circle touc... Question

A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA

Open in App Solution

Given, a quad. ABCD and a circle touches its all four sides at P,Q, R, and S respectively.

To prove: AB + CD = BC + DA

Proof:

We know that tangents drawn from external point to a circle are equal.

So, AP=AS, BP=BQ, CQ=CR, DR=DS. L.H.S. = AB + CD = AP + PB + CR + RD = AS + BQ + CQ + DS

(Tangents from same external point are always equal)

= (AS + SD) + (BQ + QC)

= AD + BC = R.H.S. Hence Proved. Suggest Corrections 13

SIMILAR QUESTIONS

**Q.**A circle touches a sides of quadrilateral ABCD prove that AB

+ CD = BC + DA.

**Q.**

If the sides of a quadrilateral ABCD touch a circle, prove that :

AB + CD = BC + AD.

**Q.**In Fig. 2, a quadrilateral ABCD is drawn to circumscribe a circle, with centre O, in such a way that the sides AB, BC, CD and DA touch the circle at the points, P, Q, R and S respectively. Prove that

A B + C D = B C + D A .

**Q.**In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose sides AB = 6cm, BC =7cm and CD =4cm.Then AD=

**Q.**

A circle touches all the four sides of a quadrilateral ABCD with AB=10cm, BC=8cm and CD=6cm.Find AD.

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