Quadrilateral Area Calculation Using Diagonal Ratios
This section details the calculation for the area of triangle BCD in a quadrilateral ABCD, given the area of triangle ABD and the ratio of diagonal segments.
Understanding Triangle Area Proportionality
The core principle used is that areas of triangles with the same height are proportional to their bases. This applies when diagonals intersect inside a quadrilateral.
Deriving the Area of Triangle BCD
Given: Quadrilateral ABCD, diagonals intersect at O. Area(\(\triangle ABD\)) = \(p\). Ratio AO : OC = m : n.
Objective: Find Area(\(\triangle BCD\)).
Step 1: Relate Areas using Diagonal Ratio
- Consider \(\triangle ABO\) and \(\triangle CBO\). They share the same height from vertex B to diagonal AC. Their area ratio equals the base ratio AO:OC.
\(\frac{\text{Area}(\triangle ABO)}{\text{Area}(\triangle CBO)} = \frac{AO}{OC} = \frac{m}{n}\)
This implies: Area(\(\triangle ABO\)) = \(\frac{m}{n} \times \text{Area}(\triangle CBO)\). - Similarly, consider \(\triangle ADO\) and \(\triangle CDO\). They share the same height from vertex D to diagonal AC. Their area ratio also equals AO:OC.
\(\frac{\text{Area}(\triangle ADO)}{\text{Area}(\triangle CDO)} = \frac{AO}{OC} = \frac{m}{n}\)
This implies: Area(\(\triangle ADO\)) = \(\frac{m}{n} \times \text{Area}(\triangle CDO)\).
Step 2: Use the Given Area of \(\triangle ABD\)
- The area of \(\triangle ABD\) is the sum of areas of \(\triangle ABO\) and \(\triangle ADO\).
\(\text{Area}(\triangle ABD) = \text{Area}(\triangle ABO) + \text{Area}(\triangle ADO) = p\)
- Substitute the relationships from Step 1 into this equation:
\(\left( \frac{m}{n} \times \text{Area}(\triangle CBO) \right) + \left( \frac{m}{n} \times \text{Area}(\triangle CDO) \right) = p\)
- Factor out \(\frac{m}{n}\):
\(\frac{m}{n} \times (\text{Area}(\triangle CBO) + \text{Area}(\triangle CDO)) = p\)
Step 3: Determine the Area of \(\triangle BCD\)
- The area of \(\triangle BCD\) is the sum of the areas of \(\triangle CBO\) and \(\triangle CDO\).
\(\text{Area}(\triangle BCD) = \text{Area}(\triangle CBO) + \text{Area}(\triangle CDO)\)
- From the equation in Step 2:
\(\frac{m}{n} \times \text{Area}(\triangle BCD) = p\)
- Solve for Area(\(\triangle BCD\)):
\(\text{Area}(\triangle BCD) = p \times \frac{n}{m} = \frac{np}{m}\)
The area of triangle BCD is \(\frac{np}{m}\).
Similar Questions
- 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 ?
- ABCD is a quadrilateral with sides AB = 9 cm, BC = 40 cm, CD = 28 cm and DA = 15 cm and one of the diagonals AC = 41 cm. Which of the following statements is/are correct ?
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 : - The perimeter of a right-angled isosceles triangle is 20 units. If \(\alpha\) is the area of the triangle, then
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- 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 $\pi = \frac{22}{7}$
