SATHEE: Circle 3 Question 7

Circle 3 Question 7

7. Let $A B C D$ be a quadrilateral with area 18 , with side $A B$ parallel to the side $C D$ and $A B = 2 C D$ . Let $A D$ be perpendicular to $A B$ and $C D$ . If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is

$(2007, 3 M)$

(a) 3

(b) 2

(d) 1

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Answer:

Correct Answer: 7. (b)

Solution:

$18 = \frac{1}{2} (3 α) (2 r)$

$\Rightarrow α r = 6$

Line, $y = - \frac{2 r}{α} (x - 2 α)$ is tangent to circle

$\begin{array}{r} (x - r)^{2} + (y - r)^{2} = r^{2} \\ 2 α = 3 r, α r = 6 and r = 2 \end{array}$

Alternate Solution

$\frac{1}{2} (x + 2 x) \times 2 r = 18$

$x r = 6$

In $△ A O B, \tan θ = \frac{x - r}{r}$ and in $△ D O C$ ,

$\begin{aligned} \tan (90^{\circ} - θ) & = \frac{2 x - r}{r} \\ ∴ & \frac{x - r}{r} & = \frac{r}{2 x - r} \\ \Rightarrow & x (2 x - 3 r) & = 0 \\ \Rightarrow & x & = \frac{3 r}{2} & \dots \end{aligned}$

From Eqs. (i) and (ii), we get

$r = 2$

Circle 3 Question 7

7. Let $A B C D$ be a quadrilateral with area 18 , with side $A B$ parallel to the side $C D$ and $A B = 2 C D$ . Let $A D$ be perpendicular to $A B$ and $C D$ . If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is

Answer:

Alternate Solution

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Circle 3 Question 7

7. Let ABCD be a quadrilateral with area 18 , with side AB parallel to the side CD and AB=2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is

Answer:

Alternate Solution

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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7. Let $A B C D$ be a quadrilateral with area 18 , with side $A B$ parallel to the side $C D$ and $A B = 2 C D$ . Let $A D$ be perpendicular to $A B$ and $C D$ . If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is