Quadrilateral Perimeter Calculation Using Tangent Properties
The problem asks for the perimeter of a quadrilateral ABCD where all sides touch a circle. Such a quadrilateral is called a tangential quadrilateral.
Key Property: A quadrilateral is tangential if and only if the sums of its opposite sides are equal. That is, for quadrilateral ABCD:
$ AB + CD = BC + DA $
Tangent Property: Tangents drawn from an external point to a circle are equal in length.
- From point A: $AP = AS$
- From point B: $BP = BQ$
- From point C: $CR = CQ$
- From point D: $DR = DS$
Applying Given Values
We are given:
- $AB = 16 \text{ cm}$
- $DS = 10 \text{ cm}$
- $RC = 3 \text{ cm}$
Using the tangent property:
- Since $DS = 10 \text{ cm}$, then $DR = 10 \text{ cm}$.
- Since $RC = 3 \text{ cm}$, then $CQ = 3 \text{ cm}$.
Calculating Side Lengths
Now we can find the lengths of CD, BC, and DA in terms of unknown tangent lengths AP, BP (or AS, BQ):
- $CD = CR + DR = 3 \text{ cm} + 10 \text{ cm} = 13 \text{ cm}$
- $BC = BQ + CQ = BQ + 3 \text{ cm}$
- $DA = DS + AS = 10 \text{ cm} + AS$
We know $AB = AP + BP$. Since $AP = AS$ and $BP = BQ$, we have:
$ AB = AS + BQ = 16 \text{ cm} $
Calculating Perimeter
The perimeter of quadrilateral ABCD is $P = AB + BC + CD + DA$.
Substitute the known and derived lengths:
$ P = 16 + (BQ + 3) + 13 + (10 + AS) $
Rearrange the terms:
$ P = (16 + 13) + (BQ + AS) + (3 + 10) $
$ P = 29 + (BQ + AS) + 13 $
We know $AS + BQ = 16 \text{ cm}$ (from $AB = 16$):
$ P = 29 + 16 + 13 $
$ P = 58 \text{ cm} $
Alternatively, using the property $AB + CD = BC + DA$:
$ 16 + 13 = (BQ + 3) + (10 + AS) $
$ 29 = 13 + BQ + AS $
$ BQ + AS = 29 - 13 = 16 \text{ cm} $
The perimeter is also $P = (AB + CD) + (BC + DA)$. Since $AB + CD = BC + DA$, the perimeter is:
$ P = 2 \times (AB + CD) = 2 \times (16 + 13) = 2 \times 29 = 58 \text{ cm} $
Or
$ P = 2 \times (BC + DA) = 2 \times ((BQ + 3) + (10 + AS)) = 2 \times (13 + BQ + AS) = 2 \times (13 + 16) = 2 \times 29 = 58 \text{ cm} $
The perimeter of the quadrilateral ABCD is 58 cm.
Similar Questions
- The length of a tangent drawn to a circle of radius 4 cm from a point 5 cm away from the center of the circle is:
- If SA and SB are tangents to a circle from an outside point S, such that SA = 12 cm and angle ASB = 60°. What will the length of chord AB be?
- In a circle, two chords of lengths x and y subtend the angles of $60^{\circ}$ and $90^{\circ}$ at the center respectively. Which of the following is correct in this scenario?
- A chord of length 24 cm is at a distance of 5 cm from the centre of a circle. The radius of the circle is ______ cm.
- ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle through A, B, C and D. If $\angle ADC = 129^\circ$, then what is the measure of $\angle BAC$?
- Two equal circles, each having a radius of $24\text{ cm}$, intersect each other, such that each passes through the centre of the other. The length of the common chord is ________ $\text{cm}$.
In the given figure, DE is tangent to the circle at a point B. $\Delta \text{ABC}$ is constructed such that A and C are points on the circumference of the given circle, with $\text{AB} = \text{AC}$. If $\angle \text{ABD} = 65^\circ$, then what is $\angle \text{BAC}$? (Figure is not to scale)
In the given circle, diameter AB is extended to meet chord CD extended at P. If the lengths of the line segments AP, CP and DP are 12 cm, 8 cm and 6 cm, respectively, what is the length of the radius of the circle?
- A point P is 13 cm from the center of a circle. The length of the tangent drawn from P to the circle is 12 cm. Find the radius of the circle.
- AB and CD are two chords of a circle. AB = 6 cm, CD = 8 cm and AB || CD. If the distance between AB and CD is 7 cm and the centre of the circle lies in the interior of the quadrilateral ABCD, then find the radius of the circle (in cm).
Important Questions from Circles, Chords and Tangents
- A circular coin of radius 1 cm is allowed to roll freely on the periphery over a circular disc of radius 10 cm. If the disc has no movement and the coin completes one revolution rolling on the periphery over the disc and without slipping, then what is the number of times the coin rotated about its centre ?
There are two concentric circles. The radii of the two circles are 100 m and 110 m respectively. A wheel of radius 30 cm rolls on the smaller circle and another wheel rolls on the larger circle. After they have completed one revolution, it is found that the two wheels rolled equal number of times
on their respective axes. What is the radius of the other wheel ?- The length of a tangent drawn to a circle of radius 4 cm from a point 5 cm away from the center of the circle is:
- If SA and SB are tangents to a circle from an outside point S, such that SA = 12 cm and angle ASB = 60°. What will the length of chord AB be?
- In a circle, two chords of lengths x and y subtend the angles of $60^{\circ}$ and $90^{\circ}$ at the center respectively. Which of the following is correct in this scenario?
