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AB is a chord of length 24 cm of a circle of radius 13 cm. The tangents at A and B intersect at a point C. Find the length AC.
- a)31.2 cm
- b)12 cm
- c)28.8 cm
- d)25 cm
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
AB is a chord of length 24 cm of a circle of radius 13 cm. The tangent...
Given, Chord AB = 24 cm, Radius OB = OA = 13 cm.
Draw OP ⊥ AB
In D OPB, OP ⊥ AB ⇒ AP = PB
[Perpendicular from centre on chord bisect the chord] =(1/2)AB= 12
Also, OB2 = OP2 + PB2
⇒ (13)2 = OP2 + PB2 ⇒ 169 = OP2 + (12)2
⇒ OP2 = 169 - 144 = 25 ⇒ OP = 5 cm
In Δ BPC, BC2 = x2 + BP2 [By pythagoras theorem]
BC2 = x2 + 144 ...(i)
In ΔOBC, OC2 = OB2 + BC2
⇒ (x + 5)2 = (13)2 + BC2 ⇒ x = 288/10=28.8cm
Put the value of x in (i), we get
BC2 = x2 + 144 = ((144)2/25) + 144⇒ BC = 31.2
⇒ AC = BC = 31.2 cm
Draw OP ⊥ AB
In D OPB, OP ⊥ AB ⇒ AP = PB
[Perpendicular from centre on chord bisect the chord] =(1/2)AB= 12
Also, OB2 = OP2 + PB2
⇒ (13)2 = OP2 + PB2 ⇒ 169 = OP2 + (12)2
⇒ OP2 = 169 - 144 = 25 ⇒ OP = 5 cm
In Δ BPC, BC2 = x2 + BP2 [By pythagoras theorem]
BC2 = x2 + 144 ...(i)
In ΔOBC, OC2 = OB2 + BC2
⇒ (x + 5)2 = (13)2 + BC2 ⇒ x = 288/10=28.8cm
Put the value of x in (i), we get
BC2 = x2 + 144 = ((144)2/25) + 144⇒ BC = 31.2
⇒ AC = BC = 31.2 cm
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Community Answer
AB is a chord of length 24 cm of a circle of radius 13 cm. The tangent...
To find the length AC, we need to apply the properties of tangents and chords in a circle.
Given:
Chord AB = 24 cm
Radius of the circle = 13 cm
Properties of Tangents and Chords:
1. The tangent to a circle is perpendicular to the radius drawn to the point of tangency.
2. Two tangents drawn from an external point to a circle are equal in length.
Let's solve the problem step by step:
Step 1: Draw the diagram
Draw a circle with center O and radius 13 cm. Mark a point C outside the circle. Draw tangents from C to the circle, which intersect the circle at points A and B.
Step 2: Identify the given information
We are given the length of chord AB, which is 24 cm, and the radius of the circle, which is 13 cm.
Step 3: Apply the properties of tangents and chords
Since the tangents from an external point to a circle are equal in length, we can conclude that CA = CB.
Step 4: Calculate the length of AC
Since CA = CB, we can consider CA as x and CB as x. Therefore, we have:
CA + AB + CB = 2x + 24 = perimeter of triangle ABC
Step 5: Calculate the perimeter of triangle ABC
The perimeter of triangle ABC can be calculated using the lengths of its sides.
We know that AB = 24 cm and the radius of the circle = 13 cm. The radius is perpendicular to the tangent at the point of tangency. Therefore, triangle ABC is a right-angled triangle.
Using Pythagoras' theorem, we can find the length of the hypotenuse (AB):
AB² = AC² + BC²
24² = x² + 13²
576 = x² + 169
x² = 576 - 169
x² = 407
x = √407
Step 6: Calculate the perimeter of triangle ABC
Using the lengths of the sides, we can calculate the perimeter of triangle ABC:
Perimeter of triangle ABC = 2x + 24
= 2(√407) + 24
Step 7: Simplify and find the value of the perimeter
Using a calculator, we can find the value of the perimeter of triangle ABC:
Perimeter of triangle ABC ≈ 31.2 cm
Therefore, the length of AC is approximately 31.2 cm.
Hence, the correct answer is option A) 31.2 cm.
Given:
Chord AB = 24 cm
Radius of the circle = 13 cm
Properties of Tangents and Chords:
1. The tangent to a circle is perpendicular to the radius drawn to the point of tangency.
2. Two tangents drawn from an external point to a circle are equal in length.
Let's solve the problem step by step:
Step 1: Draw the diagram
Draw a circle with center O and radius 13 cm. Mark a point C outside the circle. Draw tangents from C to the circle, which intersect the circle at points A and B.
Step 2: Identify the given information
We are given the length of chord AB, which is 24 cm, and the radius of the circle, which is 13 cm.
Step 3: Apply the properties of tangents and chords
Since the tangents from an external point to a circle are equal in length, we can conclude that CA = CB.
Step 4: Calculate the length of AC
Since CA = CB, we can consider CA as x and CB as x. Therefore, we have:
CA + AB + CB = 2x + 24 = perimeter of triangle ABC
Step 5: Calculate the perimeter of triangle ABC
The perimeter of triangle ABC can be calculated using the lengths of its sides.
We know that AB = 24 cm and the radius of the circle = 13 cm. The radius is perpendicular to the tangent at the point of tangency. Therefore, triangle ABC is a right-angled triangle.
Using Pythagoras' theorem, we can find the length of the hypotenuse (AB):
AB² = AC² + BC²
24² = x² + 13²
576 = x² + 169
x² = 576 - 169
x² = 407
x = √407
Step 6: Calculate the perimeter of triangle ABC
Using the lengths of the sides, we can calculate the perimeter of triangle ABC:
Perimeter of triangle ABC = 2x + 24
= 2(√407) + 24
Step 7: Simplify and find the value of the perimeter
Using a calculator, we can find the value of the perimeter of triangle ABC:
Perimeter of triangle ABC ≈ 31.2 cm
Therefore, the length of AC is approximately 31.2 cm.
Hence, the correct answer is option A) 31.2 cm.
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AB is a chord of length 24 cm of a circle of radius 13 cm. The tangents at A and B intersect at a point C. Find the length AC.a)31.2 cmb)12 cmc)28.8 cmd)25 cmCorrect answer is option 'A'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about AB is a chord of length 24 cm of a circle of radius 13 cm. The tangents at A and B intersect at a point C. Find the length AC.a)31.2 cmb)12 cmc)28.8 cmd)25 cmCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for AB is a chord of length 24 cm of a circle of radius 13 cm. The tangents at A and B intersect at a point C. Find the length AC.a)31.2 cmb)12 cmc)28.8 cmd)25 cmCorrect answer is option 'A'. Can you explain this answer?.
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