In the figure, a circle is inscribed in a quadrilateral $ A B C D $ in which $ \angle B=90^{\circ} $. If $ A D=23 \mathrm{~cm}, A B=29 \mathrm{~cm} $ and $ D S=5 \mathrm{~cm} $, find the radius $ r $ of the circle. "
Given:
In the figure, a circle is inscribed in a quadrilateral \( A B C D \) in which \( \angle B=90^{\circ} \).
\( A D=23 \mathrm{~cm}, A B=29 \mathrm{~cm} \) and \( D S=5 \mathrm{~cm} \).
To do:
We have to find the radius \( r \) of the circle.
Solution:
From the figure,
$OQ=OP=r$
$AB$ and $BC$ are tangents to the circle and $OP$ and $OQ$ are radii of the circle.
$OP\ perp\ BC$ and $OQ\ perp\ AB$
$\angle OPB = \angle OQB = 90^o$
$PBQO$ is a square.
$DS$ and $DR$ are tangents to the circle.
This implies,
$DR = DS = 5\ cm$
$AR = AD - DR$
$= 23 - 5$
$= 18\ cm$
$AR$ and $AQ$ are the tangents to the circle.
$AQ = AR = 18\ cm$
$AB = 29\ cm$
$BQ = AB - AQ$
$= 29 - 18$
$= 11\ cm$
Therefore,
Side of the square $PBQO$ is 11 cm.
This implies,
$OP = OQ = 11\ cm$
The radius of the circle is 11 cm.
Related Articles In the figure, a circle touches all the four sides of a quadrilateral \( A B C D \) with \( A B=6 \mathrm{~cm}, B C=7 \mathrm{~cm} \) and \( C D=4 \mathrm{~cm} . \) Find \( A D \)."\n
In the below figure, \( A B C D \) is a trapezium of area \( 24.5 \mathrm{~cm}^{2} . \) In it, \( A D \| B C, \angle D A B=90^{\circ} \), \( A D=10 \mathrm{~cm} \) and \( B C=4 \mathrm{~cm} \). If \( A B E \) is a quadrant of a circle, find the area of the shaded region. (Take \( \pi=22 / 7) \)."\n
In the figure, a \( \triangle A B C \) is drawn to circumscribe a circle of radius \( 4 \mathrm{~cm} \) such that the segments \( B D \) and \( D C \) are of lengths \( 8 \mathrm{~cm} \) and \( 6 \mathrm{~cm} \) respectively. Find the lengths of sides \( A B \) and \( A C \), when area of \( \triangle A B C \) is \( 84 \mathrm{~cm}^{2} \). "\n
In figure below, \( \mathrm{ABC} \) is a triangle right angled at \( \mathrm{B} \) and \( \mathrm{BD} \perp \mathrm{AC} \). If \( \mathrm{AD}=4 \mathrm{~cm} \), and \( C D=5 \mathrm{~cm} \), find \( B D \) and \( A B \)."
In the below figure, \( O A C B \) is a quadrant of a circle with centre \( O \) and radius \( 3.5 \mathrm{~cm} \). If \( O D=2 \mathrm{~cm} \), find the area of the quadrant \( O A C B \)."\n
Choose the correct answer from the given four options:It is given that \( \triangle \mathrm{ABC} \sim \triangle \mathrm{DFE}, \angle \mathrm{A}=30^{\circ}, \angle \mathrm{C}=50^{\circ}, \mathrm{AB}=5 \mathrm{~cm}, \mathrm{AC}=8 \mathrm{~cm} \) and \( D F=7.5 \mathrm{~cm} \). Then, the following is true:(A) \( \mathrm{DE}=12 \mathrm{~cm}, \angle \mathrm{F}=50^{\circ} \)(B) \( \mathrm{DE}=12 \mathrm{~cm}, \angle \mathrm{F}=100^{\circ} \)(C) \( \mathrm{EF}=12 \mathrm{~cm}, \angle \mathrm{D}=100^{\circ} \)(D) \( \mathrm{EF}=12 \mathrm{~cm}, \angle \mathrm{D}=30^{\circ} \)
From a thin metallic piece, in the shape of a trapezium \( A B C D \), in which \( A B \| C D \) and \( \angle B C D=90^{\circ} \), a quarter circle BEFC is removed. Given \( A B=B C=3.5 \) \( \mathrm{cm} \) and \( D E=2 \mathrm{~cm} \), calculate the area of the remaining piece of the metal sheet."\n
In the below figure, \( O A C B \) is a quadrant of a circle with centre \( O \) and radius \( 3.5 \mathrm{~cm} \). If \( O D=2 \mathrm{~cm} \), find the area of the shaded region."\n
In the figure below, \( A B C D \) is a trapezium with \( A B \| D C, A B=18 \mathrm{~cm}, D C=32 \mathrm{~cm} \) and the distance between \( A B \) and \( D C \) is \( 14 \mathrm{~cm} \). Circles of equal radii \( 7 \mathrm{~cm} \) with centres \( A, B, C \) and \( D \) have been drawn. Then, find the area of the shaded region of the figure. (Use \( \pi=22 / 7) \)."\n
In the figure, \( B D C \) is a tangent to the given circle at point \( D \) such that \( B D=30 \mathrm{~cm} \) and \( C D=7 \mathrm{~cm} \). The other tangents \( B E \) and \( C F \) are drawn respectively from \( B \) and \( C \) to the circle and meet when produced at \( A \) making \( B A C \) a right angle triangle. Calculate \( A F \)."\n
In a circle of radius \( 6 \mathrm{~cm} \), a chord of length \( 10 \mathrm{~cm} \) makes an angle of \( 110^{\circ} \) at the centre of the circle. Find the length of the arc \( A B \).
Find the radius of a circle whose circumference is(a) \( 22 \mathrm{~cm} \)(b) \( 17.6 \mathrm{~cm} \)(c) \( 30.8 \mathrm{~cm} \)Take \( \pi=\frac{22}{7} \) in each case.
In fig., a circle with center O is inscirbed in a quadrilateral ABCD such that, it touches the sides BC, AB, AD and CD at points P, Q, R and S respectively, if $AB=29\ cm,AD=23\ cm\ and\ \angle B=90^{o}$ and $DS=5\ cm$, then the radius of the circle $( in\ cm)$ is: $( A) \ 11$$( B) \ 18$$( C) \ 6$$( D) \ 15$"\n
In the below figure, \( A B C D \) is a rectangle with \( A B=14 \mathrm{~cm} \) and \( B C=7 \mathrm{~cm} \). Taking \( D C, B C \) and \( A D \) as diameters, three semi-circles are drawn as shown in the figure. Find the area of the shaded region."\n
In a circle of radius \( 6 \mathrm{~cm} \), a chord of length \( 10 \mathrm{~cm} \) makes an angle of \( 110^{\circ} \) at the centre of the circle. Find the area of the sector \( O A B \).
Kickstart Your Career
Get certified by completing the course
Get Started