I. The vertices A, B, C and D of the quadrilateral lie on a circle.
II. The area of triangle ACD is \(126\text{ cm}^2\).
Select the answer using the code given below :
II only
To determine which statements are correct, we need to analyze both aspects of the question regarding the quadrilateral ABCD.
- Check if the vertices A, B, C, and D lie on a circle (Statement I):
- A quadrilateral can be inscribed in a circle if and only if the opposite angles are supplementary, or according to the Ptolemy’s theorem: \( AC \cdot BD = AB \cdot CD + AD \cdot BC \).
- Given:
\( AB = 9 \text{ cm} \), \( BC = 40 \text{ cm} \), \( CD = 28 \text{ cm} \), \( DA = 15 \text{ cm} \), and \( AC = 41 \text{ cm} \). - Using Ptolemy’s theorem: \(AC \cdot BD = AB \cdot CD + AD \cdot BC\)
To find BD, rearrange and solve:
Assuming \( BD = x \),
\( 41 \cdot x = 9 \cdot 28 + 15 \cdot 40 \).
\( x = \frac{252 + 600}{41} \).
Simplifying,
\( x = \frac{852}{41} = 20.780 \text{ (approx)} \). - Since the obtained value of BD does not provide an integer, Ptolemy's theorem is not verified using this length, suggesting that ABCD is not a cyclic quadrilateral.
- Thus, Statement I is not correct.
- Calculate the area of triangle ACD (Statement II):
- To find the area of triangle ACD, we will use Heron's formula: \(s = \frac{AC + CD + DA}{2}\)
- Calculate the semi-perimeter \( s \): \(s = \frac{41 + 28 + 15}{2} = 42 \text{ cm}\)
- Now, applying Heron's formula:
- Thus, Statement II is correct.
Therefore, upon analysis, the correct answer is statement II only.
Similar Questions
- In a quadrilateral ABCD, the diagonals intersect at O. Let the area of the triangle ABD be \(p\). If AO : OC = m : n, then what is the area of the triangle BCD ?
- ABC is a triangle right-angled at C. Let P be the midpoint of BC. If \(\text{AP} = 4\sqrt{13} \text{ cm}\) and \(\text{AB} = 20 \text{ cm}\), then what is the perimeter of the triangle ABC ?
- The perimeter of a right-angled isosceles triangle is 20 units. If \(\alpha\) is the area of the triangle, then
- The minute hand of a clock is 15 cm long and sweeps an area of \(15\pi\text{ cm}^2\) on the dial of the clock. How much angle does it describe during this period ?
- A square and a rectangle have the same perimeter p and their areas differ by q units. What is the square of the difference between the length and breadth of the rectangle ?
The area of a rhombus is \(720\ \text{cm}^2\) and the sum of its diagonals is \(98\ \text{cm}\). What is the perimeter of the rhombus?
On a circular metal plate of uniform thickness, 16 holes, each of diameter 2 cm, are made. If the plate thereby has lost one-ninth of its original weight, then what is the diameter of the plate?
PQRS is a square. M is a point on PS such that PM : MS = 2 : 1 and N is a point on SR such that SN : NR = 1 : 2. If the area of triangle MQN is 10 square units, then what is the perimeter of the square?
In a quadrilateral ABCD, \(\angle ABC = 90^\circ\) and \(\angle ACD = 90^\circ\). Let \(AB = p\) units, \(BC = q\) units, \(AC = r\) units, \(CD = s\) units, \(AD = t\) units, where \(p < q < r < s < t < 15\). If p, q, r, s and t are integers, then what is the area of the quadrilateral?
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: The diagonal of a rectangle ABCD (AB > BC) is \(5\sqrt{2}\) cm. What is its perimeter?
Statement I: Length AB is an integer.
Statement II: Length BC is an integer.
Which one of the following is correct in respect of the above Question and Statements?
Important Questions from 2-D Mensuration
- The length of a rectangular plot is $(x^2 + xy + y^2)$ m and its breadth is $(x^2 - 5xy - y^2)$ m.
Find its perimeter, when $x = 1$ and $y = -1$. - The ratio between the perimeter and breadth of a rectangle is 3: 1. If the area of the rectangle is $98 \text{ cm}^2$, find the perimeter (in cm) of the rectangle.
The shorter side of a rectangle is 15 cm less than the longer side. The numerical value of its area is equal to 5 times the numerical value of its perimeter. What is the length (in cm) of its longer side?
- In a circle of radius 10.5 cm, if the angle of a sector is $\frac{2\pi}{3}$, then the perimeter of the sector is (in cm):
(Take $\pi = \frac{22}{7}$) - Find the circumference (in m) of the largest circle that can be inscribed in a rectangle whose dimensions are given as 114 m and 63 m.
Take $\pi = \frac{22}{7}$
