1. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between
their centres is 4 cm. Find the length of the common chord.
2. If two equal chords of a circle intersect within the circle, prove that the segments of
one chord are equal to corresponding segments of the other chord.
3. If two equal chords of a circle intersect within the circle, prove that the line
joining the point of intersection to the centre makes equal angles with the chords.
4. If a line intersects two concentric circles (circles
with the same centre) with centre O at A, B, C and
D, prove that AB = CD
5. Three girls Reshma, Salma and Mandip are
playing a game by standing on a circle of radius
5m drawn in a park. Reshma throws a ball to
Salma, Salma to Mandip, Mandip to Reshma. If
the distance between Reshma and Salma and
between Salma and Mandip is 6m each, what is
the distance between Reshma and Mandip?
6. A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and
David are sitting at equal distance on its boundary each having a toy telephone in
his hands to talk each other. Find the length of the string of each phone.0
their centres is 4 cm. Find the length of the common chord.
2. If two equal chords of a circle intersect within the circle, prove that the segments of
one chord are equal to corresponding segments of the other chord.
3. If two equal chords of a circle intersect within the circle, prove that the line
joining the point of intersection to the centre makes equal angles with the chords.
4. If a line intersects two concentric circles (circles
with the same centre) with centre O at A, B, C and
D, prove that AB = CD
5. Three girls Reshma, Salma and Mandip are
playing a game by standing on a circle of radius
5m drawn in a park. Reshma throws a ball to
Salma, Salma to Mandip, Mandip to Reshma. If
the distance between Reshma and Salma and
between Salma and Mandip is 6m each, what is
the distance between Reshma and Mandip?
6. A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and
David are sitting at equal distance on its boundary each having a toy telephone in
his hands to talk each other. Find the length of the string of each phone.0
Answers & Comments
Step-by-step explanation:
Let's address each problem one by one:
1. **Length of the Common Chord:**
The length of the common chord can be found using the formula:
\[2\sqrt{R_1^2 - d^2}\]
where \(R_1\) is the radius of the larger circle (5 cm), and \(d\) is the distance between the centers (4 cm).
Substituting the values:
\[2\sqrt{5^2 - 4^2} = 2\sqrt{25 - 16} = 2\sqrt{9} = 2 \times 3 = 6 \text{ cm}\]
2. **Equal Chords and Segments:**
To prove this, consider two equal chords in a circle. These chords create congruent triangles, and the line joining the midpoint of one chord to the midpoint of the other chord is the perpendicular bisector of both chords. Hence, the segments of one chord are equal to the corresponding segments of the other chord.
3. **Equal Angles with Chords:**
When two equal chords intersect within a circle, the line joining the point of intersection to the center bisects the angle formed by the two chords. Therefore, it makes equal angles with the chords.
4. **Concentric Circles and Equal Chords:**
To prove that \(AB = CD\), we can use the fact that the line joining the centers of two concentric circles is a common chord. Therefore, \(AB = CD\).
5. **Distance Between Reshma and Mandip:**
Since Reshma, Salma, and Mandip are standing on a circle, and the distance between Reshma and Salma is 6 m, and between Salma and Mandip is also 6 m, you can use the law of cosines to find the distance between Reshma and Mandip.
\[d^2 = 6^2 + 6^2 - 2 \cdot 6 \cdot 6 \cdot \cos(120^\circ)\]
Solving this, you'll find the distance \(d\) between Reshma and Mandip.
6. **Length of Toy Telephone String:**
For three boys sitting at equal distance on the boundary of a circular park, with each having a toy telephone, the length of the string for each phone can be found using the radius of the circular park.
The length of the string will be the radius of the park, which is 20 m.
If you need further assistance with any specific problem, please let me know.
Step-by-step explanation:
Therefore , PB=PC and AP=PD is proved.
4. In △OMX and △ONX,
∠OMX=∠ONX=90 ∘
OX=OX(common)
OM=ON where AB and CD are equal chords and equal chords are equidistant from the centre.
△OMX≅△ONX by RHS congruence rule.
∴∠OXM=∠OXN
i.e.,∠OXA=∠OXD
Hence proved.
5. AR=AS=1/6=3m
OR=OS=OM=5m. (Radii of the circle) In
OA²+AR²=OR²
OA¹+(3m²)=(5m)²
OA²=(25−9)m²=16m²
OA=4m
ORSM will be a kite (OR = OM and RS = SM). We know that diagonals of a kite are perpendicular and the diagonal common to both the isosceles traingle is bisected by another diagonal
∴∠RCS will be of 90∘ and RC=CM
Area of ΔORS=1/2×OA×RS
1/2×RC×OS=1/2×4×6
RC×5=24
RC=4.8
RM=2RC=2(4.8)=9.6
Therefore, the distance between Reshma and Mandip is 9.6 m.