The diagonals of a cyclic quadrilateral ABCD intersect at P and the area of the triangle APB is 24 square cm. If AB = 8 cm and CD = 5 cm, then what is the area of the circle CPD?
9.375 square cm
Solving the Cyclic Quadrilateral Problem: Finding Triangle Area
This problem involves a cyclic quadrilateral, its diagonals, and the areas of triangles formed by their intersection. We are given the area of one such triangle and the lengths of two opposite sides of the quadrilateral, and we need to find the area of another triangle formed by the intersecting diagonals.
Understanding Cyclic Quadrilaterals and Intersecting Diagonals
A cyclic quadrilateral is a four-sided figure whose vertices all lie on a single circle. When the diagonals of a cyclic quadrilateral intersect inside the circle, they form four triangles. In our case, the diagonals AC and BD of the cyclic quadrilateral ABCD intersect at point P, forming triangles APB, BPC, CPD, and DPA.
A key property related to the intersecting diagonals in a cyclic quadrilateral is the similarity of certain pairs of triangles:
- Triangle APB is similar to Triangle CPD (\(\triangle \text{APB} \sim \triangle \text{CPD}\))
- Triangle APD is similar to Triangle BPC (\(\triangle \text{APD} \sim \triangle \text{BPC}\))
Let's focus on the first pair, \(\triangle \text{APB}\) and \(\triangle \text{CPD}\). These triangles are similar due to the properties of angles subtended by the same arc in a circle:
- \(\angle \text{PAB} = \angle \text{PDC}\) (Angles subtended by the same arc BC)
- \(\angle \text{PBA} = \angle \text{PCD}\) (Angles subtended by the same arc AD)
- \(\angle \text{APB} = \angle \text{CPD}\) (Vertically opposite angles)
Since all three corresponding angles are equal, \(\triangle \text{APB}\) is similar to \(\triangle \text{CPD}\) by the AAA similarity criterion.
Using the Ratio of Areas of Similar Triangles
A fundamental theorem states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. For similar triangles \(\triangle \text{APB}\) and \(\triangle \text{CPD}\), the corresponding sides opposite the equal angles are AB and CD, AP and CP, and BP and DP.
Therefore, we can write the relationship between their areas and corresponding sides as:
\begin{equation*} \frac{\text{Area}(\triangle \text{APB})}{\text{Area}(\triangle \text{CPD})} = \left(\frac{\text{AB}}{\text{CD}}\right)^2 = \left(\frac{\text{AP}}{\text{CP}}\right)^2 = \left(\frac{\text{BP}}{\text{DP}}\right)^2 \end{equation*}
Calculating the Area of Triangle CPD
We are given:
- Area(\(\triangle\) APB) = 24 square cm
- AB = 8 cm
- CD = 5 cm
We want to find Area(\(\triangle\) CPD). Using the ratio of areas formula with the given side lengths AB and CD (which are corresponding sides in similar triangles APB and CPD, opposite to \(\angle\)APB and \(\angle\)CPD respectively):
\begin{equation*} \frac{\text{Area}(\triangle \text{APB})}{\text{Area}(\triangle \text{CPD})} = \left(\frac{\text{AB}}{\text{CD}}\right)^2 \end{equation*}
Substitute the given values into the equation:
\begin{equation*} \frac{24}{\text{Area}(\triangle \text{CPD})} = \left(\frac{8}{5}\right)^2 \end{equation*}
Calculate the square of the ratio of the sides:
\begin{equation*} \left(\frac{8}{5}\right)^2 = \frac{8^2}{5^2} = \frac{64}{25} \end{equation*}
So, the equation becomes:
\begin{equation*} \frac{24}{\text{Area}(\triangle \text{CPD})} = \frac{64}{25} \end{equation*}
Now, solve for Area(\(\triangle\) CPD):
\begin{equation*} \text{Area}(\triangle \text{CPD}) = 24 \times \frac{25}{64} \end{equation*}
Perform the multiplication:
\begin{equation*} \text{Area}(\triangle \text{CPD}) = \frac{24 \times 25}{64} = \frac{600}{64} \end{equation*}
Simplify the fraction:
\begin{equation*} \frac{600}{64} = \frac{300}{32} = \frac{150}{16} = \frac{75}{8} \end{equation*}
Convert the fraction to a decimal:
\begin{equation*} \frac{75}{8} = 9.375 \end{equation*}
Thus, the area of triangle CPD is 9.375 square cm.
Summary of Calculation
| Quantity | Value |
|---|---|
| Area(\(\triangle\) APB) | 24 sq cm |
| AB | 8 cm |
| CD | 5 cm |
| Ratio of sides (\(\frac{\text{AB}}{\text{CD}}\)) | \(\frac{8}{5}\) |
| Square of ratio of sides (\(\left(\frac{\text{AB}}{\text{CD}}\right)^2\)) | \(\frac{64}{25}\) |
| Area(\(\triangle\) CPD) | \(24 \times \frac{25}{64} = 9.375\) sq cm |
The area of triangle CPD is 9.375 square cm.
Revision Table: Key Concepts for Cyclic Quadrilaterals
| Concept | Description | Relevance to Problem |
|---|---|---|
| Cyclic Quadrilateral | A quadrilateral whose vertices lie on a circle. | Property used to establish similarity of triangles formed by diagonals. |
| Angles Subtended by Same Arc | Angles subtended by the same arc at the circumference are equal. | Used to prove \(\triangle\) APB \(\sim\) \(\triangle\) CPD. |
| Vertically Opposite Angles | Angles formed by two intersecting lines that are opposite to each other at the intersection point. They are equal. | Used to prove \(\triangle\) APB \(\sim\) \(\triangle\) CPD (\(\angle\)APB = \(\angle\)CPD). |
| Similar Triangles | Triangles with the same shape but possibly different sizes. Corresponding angles are equal, and corresponding sides are proportional. | \(\triangle\) APB and \(\triangle\) CPD are similar. |
| Ratio of Areas of Similar Triangles | The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. | The primary formula used to solve the problem. |
Additional Information: Properties of Cyclic Quadrilaterals
Beyond the triangle similarity used in this problem, cyclic quadrilaterals have other important properties:
- Opposite Angles are Supplementary: The sum of opposite angles in a cyclic quadrilateral is 180 degrees (\(\angle\)A + \(\angle\)C = 180°, \(\angle\)B + \(\angle\)D = 180°).
- Ptolemy's Theorem: For a cyclic quadrilateral ABCD, the product of the lengths of the diagonals is equal to the sum of the products of the lengths of the opposite sides: AC \(\times\) BD = (AB \(\times\) CD) + (BC \(\times\) AD).
- Power of a Point Theorem: The intersection point P of the diagonals has the property that AP \(\times\) PC = BP \(\times\) PD. This relates the segments of the diagonals.
- Angles in the Same Segment: Angles subtended by the same chord in the same segment of the circle are equal (as used for \(\angle\)PAB = \(\angle\)PDC and \(\angle\)PBA = \(\angle\)PCD).
These properties are useful in solving various geometry problems involving circles and quadrilaterals.
Similar Questions
The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?
Segment QR of length r is a tangent at Q to a circle of radius r with centre at P. What is the area of the part of the triangle PQR, which is outside the circular region?
Consider the following statements:
I. \(\angle ABC\) lies between 60° and 90°.
II. If z is the distance in cm from the centre O of the circle to the midpoint of BC, then \(7\text{ cm} < z < 8\text{ cm}\).
Which of the statements given above is/are correct?
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: ABCD is a quadrilateral with \(AB = 6\) cm, \(BC = 8\) cm, \(CD = 4\sqrt{5}\) cm, \(DA = 2\sqrt{5}\) cm and inscribed in a circle. What is the diameter of the circle?
Statement I: The area of the quadrilateral is \(44\) cm².
Statement II: \(\angle ABC = 90^\circ\).
Which one of the following is correct in respect of the above Question and Statements?
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: ABC is a triangle inscribed in a semi-circle such that AC coincides with the diameter of the semi-circle. Let P be any point on the arc of the semi-circle. What is \(AP^2 + CP^2 + AC^2\) equal to?
Statement I: The radius of the circle is 5 cm.
Statement II: \(AB = 6\text{ cm}\) and \(BC = 8\text{ cm}\).
Which one of the following is correct in respect of the above Question and Statements?
What is \(AD : DB\) equal to?
If x is the area of the circle and y is the area of the triangle such that \(z = \left(\frac{x}{y}\right)^2\), then which one of the following is correct?
What is the ratio of the area of the triangle ADO to the area of the quadrilateral BDOM?
Important Questions from Circles
The sum of the radius and diameter of a circle is 84 cm. What is the circumference of this circle?
The tangent at a point C of a circle and diameter AB when extended intersect at D, if ∠DCA = 110°, then ∠CBA is equal to
The equation of a circle with diameters are 2x - 3y + 12 = 0 and x + 4y - 5 = 0 and area of 154 sq. units is
The internal center of similitude of two circles $ (x - 1)^2 + (y - 3)^2 = 4 $ and $ (x + 5)^2 + (y - 9)^2 = 16 $ is
The inner circumference of a circular race track 14 cm wide is 440 cm. Find the radius of the outer circle.
