I. Area of triangle OAP is equal to area of triangle OCQ.
II. Area of triangle OAC is equal to \(102 \text{ cm}^2\).
Select the answer using the code given below :
I only
To solve this question, we need to examine the geometric properties of the circle and its chords. Let's break down the statements one by one:
- **Statement I: Area of triangle OAP is equal to area of triangle OCQ.**
- The chords AB and CD are parallel and on opposite sides of the center O. Let P and Q be the midpoints of AB and CD respectively.
- Because the chords are parallel and the distance between them is constant, triangles OAP and OCQ will be similar by the method of congruent altitudes from point O to the line segments AB and CD. Therefore, the areas of triangles OAP and OCQ will be equal due to symmetry about the center O.
- Hence, Statement I is correct.
- **Statement II: Area of triangle OAC is equal to 102 cm².**
- To verify this, consider the circle's symmetry and geometry:
- Using the Pythagorean theorem, let x be the distance from O to chord AB, and y as the distance from O to chord CD.
- Since AB = 10 cm and CD = 24 cm, and the distance between the chords is 17 cm, from the midpoints, OP = x and OQ = y, we find the coordinates of P and Q to solve for x and y.
- Using properties of perpendicular bisectors and Pythagorean theorem, we assess the potential for areas based on known geometric distances.
- However, without the radius or further data, specifically the exact or relative positions of A and C on the circle, the calculation of the exact area to verify the statement numerically is challenging without assuming standard geometric properties. Therefore, based on this insight without further details, the statement's validity as a certain 102 cm² is questionable without additional assumptions or data checks.
Therefore, based on the geometry outlined and lack of complete certainty for Statement II without additional data points, Statement I only is correct. The correct option based on this reasoning is "I only".
Important Questions from Geometry
The sides of a triangle are in the ratio 6 : 4 : 3 and its perimeter is 104 cm. The length of the longest side (in cm) is:
An isosceles right-angled triangle has hypotenuse length as 10 units. What is the area of the triangle (in square units)?
Two circles of radii 16 cm and 4 cm, respectively, touch each other externally at Point A. PQ is the direct common tangent of these circles with centres C1 and C2, respectively. What is the length of PQ?
Let C be a circle with center O and AB be a chord of C such that the length of AB is equal to the radius of C. Let D be any point on the major arc of AB. Find ∠AOB and ∠ADB, respectively.
The centres of two circles are 84 cm apart. If the radii of these two circles are 38 cm and 26 cm, respectively, then which of the following options gives the length (in cm) of a direct common tangent of these two circles?
