Question

ABCD is a trapezium in which AB is parallel to DC, AD is perpendicular to AB, and $AB = 3DC$. If a circle inscribed in the trapezium touching all the sides has a radius of 3 cm, then the area, in sq. cm, of the trapezium is

• A. \(54\)
• B. \(30\sqrt{3}\)
• C. \(48\)
• D. \(36\sqrt{2}\)

Hint:

For figures with an incircle:
The distance between two parallel tangents equals twice the radius.
In tangential quadrilaterals (those with an incircle), using coordinates with the incenter at convenient positions (like \((r, r)\)) can simplify distance calculations.

Answer 1

The Correct Option is C

Approach: A circle inscribed in a right trapezium pins down the height immediately (height \(=\) diameter), and the tangential-polygon rule (sum of opposite sides equal) gives the slant side without coordinates.

Step 1: Get the height. The incircle touches both parallels \(AB\) and \(DC\), so the gap between them is the diameter: \(AD = 2r = 6\) cm. (Here \(AD\perp AB\), so \(AD\) IS the height.) Let \(DC = c\), \(AB = 3c\).

Step 2: Use the tangential property. For any quadrilateral with an inscribed circle, the two pairs of opposite sides have equal sums:
\[ AB + DC = AD + BC. \]
So \(3c + c = 6 + BC \Rightarrow BC = 4c - 6\).

Step 3: Pin BC with the right angle at A and D. Drop the horizontal/vertical: the slant side \(BC\) spans a vertical drop of \(6\) (the height) and a horizontal run of \(AB - DC = 3c - c = 2c\). By Pythagoras:
\[ BC^2 = 6^2 + (2c)^2 = 36 + 4c^2. \]

Step 4: Solve for \(c\). Equate the two expressions for \(BC\):
\[ (4c - 6)^2 = 36 + 4c^2 \Rightarrow 16c^2 - 48c + 36 = 36 + 4c^2 \Rightarrow 12c^2 = 48c \Rightarrow c = 4. \]
So \(DC = 4\), \(AB = 12\), \(BC = 4(4) - 6 = 10\).

Intuition: The "equal sums of opposite sides" rule is the fastest handle on any inscribed-circle quadrilateral — reach for it before coordinates.

Step 5: Area. \[ \text{Area} = \tfrac12(AB + DC)\times AD = \tfrac12(12 + 4)\times 6 = \boxed{48}\ \text{cm}^2, \] matching option (c).

Answer 2

Let the trapezium \(ABCD\) have \(AB \parallel DC\), \(AD \perp AB\), and let \[ AB = a,\quad DC = c,\quad \text{with } a = 3c. \] The radius of the incircle is \(r = 3\) cm. 
Step 1: Use the fact that the circle touches both parallel sides. Since the circle is tangent to both \(AB\) and \(DC\), the perpendicular distance between these two parallel lines is equal to the diameter of the circle: \[ \text{height} = AD = 2r = 2 \times 3 = 6 \text{ cm}. \] 
Step 2: Set up a coordinate system. Place the trapezium as follows: \[ A = (0,0), \quad B = (a,0), \quad D = (0,6), \quad C = (c,6), \] with \(a = 3c\). The circle touches: 
\(AB\) at distance 3 \(\Rightarrow\) center is 3 units above it, 
\(DC\) at distance 3 \(\Rightarrow\) center is 3 units below it, 
\(AD\) at distance 3 \(\Rightarrow\) center is 3 units to the right of it.
Hence, the center of the circle is \[ O = (3,3). \] 
Step 3: Use the distance from the center to side \(BC\). Line \(BC\) passes through \(B(a,0)\) and \(C(c,6)\). The equation of line \(BC\) is: \[ \text{slope} = \frac{6 - 0}{c - a} = \frac{6}{c - a}, \] so \[ y = \frac{6}{c - a}(x - a). \] Rewriting in standard form: \[ (c - a)y = 6x - 6a \quad \Rightarrow \quad 6x - (c - a)y - 6a = 0. \] The perpendicular distance from \(O(3,3)\) to line \(BC\) must be equal to the radius \(3\): \[ \frac{\left|6\cdot 3 - (c - a)\cdot 3 - 6a\right|}{\sqrt{6^2 + (-(c - a))^2}} = 3. \] Simplify the numerator: \[ 6\cdot 3 - 3(c - a) - 6a = 18 - 3c + 3a - 6a = 18 - 3c - 3a = 3(6 - c - a). \] So, \[ \frac{3|6 - c - a|}{\sqrt{36 + (c - a)^2}} = 3 \quad \Rightarrow \quad \frac{|6 - c - a|}{\sqrt{36 + (c - a)^2}} = 1. \] 
Step 4: Use the relation \(a = 3c\). Since \(a = 3c\), \[ a + c = 3c + c = 4c, \quad c - a = c - 3c = -2c. \] Then \[ |6 - (a + c)| = |6 - 4c|, \quad \sqrt{36 + (c - a)^2} = \sqrt{36 + (-2c)^2} = \sqrt{36 + 4c^2}. \] From \[ \frac{|6 - 4c|}{\sqrt{36 + 4c^2}} = 1 \quad \Rightarrow \quad (6 - 4c)^2 = 36 + 4c^2. \] Expanding: \[ 36 - 48c + 16c^2 = 36 + 4c^2 \Rightarrow 16c^2 - 48c = 4c^2 \Rightarrow 12c^2 - 48c = 0 \Rightarrow 12c(c - 4) = 0. \] So \(c = 4\) (since \(c>0\)), and hence \[ a = 3c = 12. \] 
Step 5: Find the area of the trapezium. The area of a trapezium is \[ \text{Area} = \frac{1}{2}(AB + DC)\cdot \text{height} = \frac{1}{2}(a + c)\cdot AD = \frac{1}{2}(12 + 4)\cdot 6 = \frac{1}{2} \times 16 \times 6 = 48. \] So, the area of the trapezium is \(48\ \text{sq. cm}\).