Circles — Andhra Pradesh (SSC) Class 9 Mathematics Solutions (Free)
Free step-by-step Andhra Pradesh (SSC) Class 9 Mathematics solutions for "Circles" — important questions with detailed answers, download PDF for board exam preparation.
TL;DR: Free step-by-step Andhra Pradesh (SSC) Class 9 Mathematics solutions for "Circles" — important questions with detailed answers, download PDF for board…
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Q1: Define a circle and state the following properties: (1) All radii of a circle are equal. (2) A chord is a line segment joining any two points on the circle. (3) The longest chord is the diameter. If a circle has radius 7 cm, find the diameter and the circumference.
Step 1: Definition of Circle: A circle is the locus of all points in a plane that are at a constant distance (radius) from a fixed point (center).
Step 2: Properties:
(1) All radii of a circle are equal: Since all points on the circle are equidistant from the center, all radii r are equal.
(2) Chord: A line segment with both endpoints on the circle.
Examples: diameter (through center), any other line segment between two points on circle.
(3) Diameter is the longest chord: Diameter = 2r (passes …
Q2: State the theorem about angles subtended by an arc at the center and on the circumference: If an arc AB subtends an angle of 60° at the center O, what angle does the same arc subtend at a point C on the circumference (on the major arc)?
Step 1: Theorem: The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circle (on the major arc).
Angle at center = 2 × Angle at circumference
Step 2: Given:
Arc AB subtends angle AOB = 60° at center O
Point C is on the major arc AB
Find: angle ACB (angle at circumference)
Step 3: Apply the theorem:
angle AOB = 2 × angle ACB
60° = 2 × angle ACB
angle ACB = 60°/2
angle ACB = 30°
Step 4: Verification: This theorem holds for all…
Q3: In circle with center O and radius 5 cm, a chord AB is at a perpendicular distance of 3 cm from the center O. Find the length of chord AB.
Step 1: Given:
Circle with center O and radius r = 5 cm
Chord AB at perpendicular distance d = 3 cm from center O
Step 2: Let M be the foot of the perpendicular from O to chord AB.
Then OM ⊥ AB and OM = 3 cm
Step 3: Important property: The perpendicular from the center of a circle to a chord bisects the chord.
Therefore: AM = MB = AB/2
Step 4: In right triangle OMA:
angle OMA = 90°
OA = 5 cm (radius, hypotenuse)
OM = 3 cm (perpendicular distance)
AM = ? (half of chord)
Step 5: By Pythagoras …
Q4: State the theorem about angles in the same segment: Two angles subtended by the same chord at two different points on the same arc are equal. In circle with center O, chord AB subtends angle 40° at point C on the major arc. Find the angle it subtends at point D, also on the major arc.
Step 1: Theorem (Angles in the Same Segment): Angles subtended by the same chord at different points on the same arc of a circle are equal.
If chord AB subtends angles at points C and D on the same arc, then angle ACB = angle ADB.
Step 2: Given:
Circle with center O
Chord AB
Point C on major arc: angle ACB = 40°
Point D also on major arc
Find: angle ADB
Step 3: Since C and D are both on the major arc (same segment), and they are both subtended by the same chord AB:
angle ADB = angle ACB
Step…
Q5: In a circle, two chords AB and CD intersect at point P inside the circle. If AP = 6 cm, PB = 4 cm, and CP = 3 cm, find the length of PD.
Step 1: Theorem (Intersecting Chords): When two chords intersect inside a circle, the products of their segments are equal.
If chords AB and CD intersect at P, then: AP × PB = CP × PD
Step 2: Given:
Chord AB with intersection point P
AP = 6 cm
PB = 4 cm
Chord CD with intersection point P
CP = 3 cm
PD = ?
Step 3: Apply the intersecting chords theorem:
AP × PB = CP × PD
6 × 4 = 3 × PD
24 = 3 × PD
Step 4: Solve for PD:
PD = 24/3
PD = 8 cm
Step 5: Verification:
Product of segments of chord AB =…
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