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Chapter Analysis
Intermediate16 pages • EnglishQuick Summary
This chapter focuses on the properties and theorems related to circles. It introduces concepts such as angles subtended by chords, equal chords and their distances from the center, and cyclic quadrilaterals. It also covers important theorems, including those related to congruent chords and cyclic parallelograms, providing mathematical proofs and examples to illustrate each point.
Key Topics
- •Circle and its parts
- •Angles subtended by chords
- •Equal chords and distances from the center
- •Cyclic quadrilaterals
- •Cyclic parallelograms
- •Angles in the same segment
- •Theorems on circles
- •Properties of cyclic figures
Learning Objectives
- ✓Understand the definitions and parts of a circle.
- ✓Explore the properties of angles subtended by chords.
- ✓Demonstrate the relationship between chords and their distances from the center.
- ✓Describe and prove properties of cyclic quadrilaterals.
- ✓Apply theorems related to circles in problem-solving.
- ✓Verify the properties of cyclic figures using visual and numerical methods.
Questions in Chapter
In Fig. 9.23, A, B, and C are three points on a circle with centre O such that ∠ BOC = 30° and ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.
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A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.
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In Fig. 9.24, ∠ PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠ OPR.
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In Fig. 9.25, ∠ ABC = 69°, ∠ ACB = 31°, find ∠BDC.
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In Fig. 9.26, A, B, C, and D are four points on a circle. AC and BD intersect at a point E such that ∠ BEC = 130° and ∠ ECD = 20°. Find ∠ BAC.
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ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠ DBC = 70°, ∠ BAC is 30°, find ∠ BCD. Further, if AB = BC, find ∠ ECD.
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If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
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If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
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Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Fig. 9.27). Prove that ∠ ACP = ∠ QCD.
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If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lies on the third side.
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ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠ CBD.
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Prove that a cyclic parallelogram is a rectangle.
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Additional Practice Questions
What is the relationship between the angles subtended by equal chords at the center of a circle?
easyAnswer: Equal chords of a circle subtend equal angles at the center.
Explain why the perpendicular from the center of a circle to a chord bisects the chord.
mediumAnswer: The perpendicular from the center of a circle to a chord bisects the chord because triangles formed by the radii and the chord are congruent by the SSS theorem.
How can you determine if a quadrilateral is cyclic?
mediumAnswer: A quadrilateral is cyclic if the sum of its opposite angles is 180°.
Why do congruent arcs subtend equal angles at the center of a circle?
easyAnswer: Congruent arcs subtend equal angles at the center because they have equal chord lengths, and thus the sectors are congruent.
Why is the angle in a semicircle a right angle?
hardAnswer: The angle in a semicircle is a right angle because it is subtended by a diameter which forms a triangle with the circle’s boundary, making it a right triangle.