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  and  respectively.
Sol:
Let ∠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
Similarly
and

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:

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  and
In ΔOMP, ∠M = 90°
By Pythagoras Theorem, we have
PM² = OP² - OM²

Again in  ΔOLP, ∠L = 90°
By Pythagoras Theorem, we have
PL² = OP² - OL²

Also, PQ = 2PR
PQ² = 4PR²   ......(iii)
From (i), (ii) and (iii) we have


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

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 ?
Sol:
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°.
∠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.