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Class 9 Maths Chapter 10 HOTS Questions - Circles

Q1: Bisectors of angles A, B and C of triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the ∠DEF are Class 9 Maths Chapter 10 HOTS Questions - Circles and Class 9 Maths Chapter 10 HOTS Questions - Circles respectively.
Sol:

Class 9 Maths Chapter 10 HOTS Questions - CirclesLet ∠BAD = x, ∠ABE = y
and ∠ACF = 2, then
∠CAD = x, ∠CBE = y
and ∠BCF = 2 [AD, BE and CF is bisector of ∠A, ∠B and ∠C]
In ∆BC,
∠A + ∠B + ∠C = 180°
⇒ 2x + 2y + 2Z = 180°
or x + y + Z = 90° …(i)
Now, ∠ADE = ∠ABE
and ∠ADF = ∠ACF [angles in the same segment of a circle]
⇒ ∠ADE = y and ∠ADF = Z
⇒ ∠ADE + ∠ADF = y + Z
or ∠D = y + Z …(ii)
From (i) and (ii), we have
x + 2D = 90°
⇒ ∠D = 90° – x
or Class 9 Maths Chapter 10 HOTS Questions - Circles
Similarly Class 9 Maths Chapter 10 HOTS Questions - Circles
and Class 9 Maths Chapter 10 HOTS Questions - Circles

Q2: PQ and PR are the two chords of a circle of radius r. If the perpendiculars drawn from the centre of the circle to these chords are of lengths a and b, PQ = 2PR, then prove that:
Class 9 Maths Chapter 10 HOTS Questions - Circles
Sol: 
In circle Clo, r), PQ and PR are two chords, draw OM I PQ, OL I PR, such that OM = a and OL = b. Join OP. Since the perpendicular from the centre of the circle to the chord of the circle, bisects the chord.
We have Class 9 Maths Chapter 10 HOTS Questions - Circles and Class 9 Maths Chapter 10 HOTS Questions - Circles
Class 9 Maths Chapter 10 HOTS Questions - CirclesIn ΔOMP, ∠M = 90°
By Pythagoras Theorem, we have
PM2 = OP2 - OM2
Class 9 Maths Chapter 10 HOTS Questions - Circles
Again in  ΔOLP, ∠L = 90°
By Pythagoras Theorem, we have
PL2 = OP2 - OL2
Class 9 Maths Chapter 10 HOTS Questions - Circles
Also, PQ = 2PR
PQ2 = 4PR2   ......(iii)
From (i), (ii) and (iii) we have
Class 9 Maths Chapter 10 HOTS Questions - Circles

Q3: A circular park of radius 10 m is situated in a colony. Three students Ashok, Raman and Kanaihya are standing at equal distances on its circumference each having a toy telephone in his hands to talk each other about Honesty, Peace and Discipline.
(i) Find the length of the string of each phone.
(ii) Write the role of discipline in students’ life.
Sol:
(i)
Let us assume A, B and C be the positions of three students Ashok, Raman and Kanaihya respectively on the circumference of the circular park with centre O and radius 10 m. Since the centre of circle coincides with the centroid of the equilateral ∆ABC.
Radius of circumscribed circle = 2/3AD
⇒ 10 = 2/3AD
⇒ AD = 15 m
Class 9 Maths Chapter 10 HOTS Questions - Circles
Thus, the length of each string is 10√3 m.
(ii) In students’ life, discipline is necessary. It motivates as well as nurture the students to make him a responsible citizen.

Q4: A small cottage industry employing people from a nearby slum area prepares round table cloths having six equal designs in the six segment formed by equal chords AB, BC, CD, DE, EF and FA. If O is the centre of round table cloth (see figure). Find ∠AOB, ∠AEB and ∠AFB. What value is depicted through this question ?
Class 9 Maths Chapter 10 HOTS Questions - CirclesSol:

Since six equal designs in the six segment formed by equal chords AB, BC, CD, DE, EF and FA.
Therefore, we have six equilateral triangles as shown in the figure. Since ∆AOB, ∆BOC, ∆COD, ∆DOE, ∆EOF
∴ Each angle is equal to 60°.
Class 9 Maths Chapter 10 HOTS Questions - Circles∠AOB = 60°
∠AOB, ∠AEB and ∠AFB are angles subtended by an arc AB at the FK centre and at the remaining part of the circle.
∴ ∠AEB = ∠AFB = 1/2 ∠AOB = 1/2 × 60° = 30°
Thus, ∠AEB = ∠AFB = 30°
Value depicted : By employing people from a slum area to prepare round table clothes realize their social responsibility to work for helping the ones in need.

The document Class 9 Maths Chapter 10 HOTS Questions - Circles is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on Class 9 Maths Chapter 10 HOTS Questions - Circles

1. What is a circle in mathematics?
Answer: In mathematics, a circle is a closed curve that consists of all points in a plane that are equidistant from a fixed point called the center. It is defined by its radius, which is the distance between the center and any point on the circle.
2. How do you find the circumference of a circle?
Answer: The circumference of a circle can be found by using the formula C = 2πr, where C represents the circumference and r is the radius of the circle. Pi (π) is a mathematical constant approximately equal to 3.14159.
3. What is the difference between the radius and diameter of a circle?
Answer: The radius of a circle is the distance from its center to any point on the circle, while the diameter is the distance across the circle passing through the center. The diameter is always twice the length of the radius.
4. How do you find the area of a circle?
Answer: The area of a circle can be calculated using the formula A = πr^2, where A represents the area and r is the radius of the circle. Pi (π) is a mathematical constant that is approximately equal to 3.14159.
5. Can you find the radius of a circle if you know its circumference?
Answer: Yes, the radius of a circle can be found if you know its circumference. The formula to calculate the radius is r = C / (2π), where r is the radius and C is the circumference. By substituting the known circumference value into the formula, you can determine the radius of the circle.
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