Circles — Karnataka (SSLC) Class 10 Mathematics Solutions (Free)
Free step-by-step Karnataka (SSLC) Class 10 Mathematics solutions for "Circles" — important questions with detailed answers, download PDF for board exam preparation.
TL;DR: Free step-by-step Karnataka (SSLC) Class 10 Mathematics solutions for "Circles" — important questions with detailed answers, download PDF for board ex…
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Q1: State and prove the theorem: A tangent to a circle is perpendicular to the radius at the point of tangency.
Theorem: A tangent to a circle is perpendicular to the radius at the point of tangency.
Given:
Let O be the center of a circle.
Let AB be a tangent to the circle at point P.
OP is the radius to point P.
Prove: OP ⊥ AB (that is, OP is perpendicular to AB)
Proof:
Step 1: Assume, for contradiction, that OP is not perpendicular to AB.
Suppose OP makes an angle other than 90° with the tangent AB.
Step 2: From O, draw a perpendicular to the tangent AB, meeting it at point Q.
Since this perpendicu…
Q2: Two tangents are drawn to a circle from an external point. Prove that they are equal in length.
Theorem: Two tangents drawn to a circle from an external point are equal in length.
Given:
Let O be the center of the circle.
Let P be an external point.
PA and PB are two tangents to the circle at points A and B respectively.
Prove: PA = PB
Proof:
Step 1: Since PA and PB are tangents, we use the perpendicularity property.
OA ⊥ PA (radius is perpendicular to tangent at point of contact)
OB ⊥ PB (radius is perpendicular to tangent at point of contact)
Step 2: Consider triangles OAP and OBP.
…
Q3: A circle with center O and radius 5 cm has a point P at distance 13 cm from the center. From P, two tangents are drawn to the circle. Find the length of each tangent.
Given:
Circle with center O and radius r = 5 cm
Point P at distance OP = 13 cm from center
Two tangents PA and PB are drawn from P to the circle
Find: Length of each tangent (PA and PB)
Diagram (text description):
A (point of tangency)
/|
/ |
/ | 5 cm (radius OA)
/ |
/ |
/ O________
/ 13 cm
/
P \_____ B (point of tangency)
Solution:
Step 1: Since PA is a tangent at point A:
OA ⊥ PA
Therefore, triangle OAP …
Q4: A chord of a circle is 24 cm long. The radius of the circle is 13 cm. Find the distance of the chord from the center of the circle.
Given:
Length of chord = 24 cm
Radius of circle = 13 cm
Find: Distance of chord from the center
Diagram (text description):
O (center)
|
| d (perpendicular distance)
|
----M---- (M is foot of perpendicular, midpoint of chord)
/ | \
A | B
|--12-|-12--| (half-chord on each side)
\__________/
Chord AB = 24 cm
Key Property:
The perpendicular from the center of a circle to a chord bisects the chord.
Step 1: Let O be the center and AB …
Q5: Two circles with centers O₁ and O₂ and radii 8 cm and 5 cm respectively touch each other externally. Find the distance between their centers.
Given:
Circle 1: Center O₁, radius r₁ = 8 cm
Circle 2: Center O₂, radius r₂ = 5 cm
The circles touch each other externally
Find: Distance between centers O₁O₂
Diagram (text description):
Circle 1 Point of tangency Circle 2
* * *
\ / \ /
\ 8 cm / \ 5 cm /
\ O₁ O₂ /
*-----------•---------•-----------*
8 cm ? 5 cm
Definition:
Two c…
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