Draw 6 Circles in a 5 Group

Balbharti Maharashtra State Board Class 9 Maths Solutions covers the Problem Set 6 Geometry 9th Class Maths Office 2 Answers Solutions Chapter six Circumvolve.

Problem Ready 6 Geometry 9th Std Maths Part 2 Answers Affiliate 6 Circle

Question one.
Choose correct alternative reply and fill up in the blanks.

i. Radius of a circle is 10 cm and altitude of a chord from the center is half-dozen cm. Hence, the length of the chord is ____.
(A) xvi cm
(B) viii cm
(C) 12 cm
(D) 32 cm
Reply:

∴ OA^two = Air conditioning² + OCⁱⁱ
∴ 10² = ACⁱⁱ + vi²
∴ AC² = 64
∴ AC = 8 cm
∴ AB = 2(AC)= 16 cm
(A) 16 cm

two. The signal of concurrence of all angle bisectors of a triangle is called the ____.
(A) centroid
(B) circumcentre
(C) incentre
(D) orthocentre
Respond:
(C) incentre

iii. The circle which passes through all the vertices of a triangle is called ____.
(A) circumcircle
(B) incircle
(C) congruent circle
(D) concentric circle
Respond:
(A) circumcircle

4. Length of a chord of a circumvolve is 24 cm. If altitude of the chord from the middle is v cm, then the radius of that circle is ____.
(A) 12 cm
(B) 13 cm
(C) 14 cm
(D) 15 cm
Answer:

OA² = AC^two + OC²
∴ OA² = 12² + v²
∴ OAⁱⁱ = 169
∴ OA = xiii cm
(B) 13 cm

v. The length of the longest chord of the circumvolve with radius 2.nine cm is ____.
(A) iii.5 cm
(B) vii cm
(C) 10 cm
(D) 5.8 cm
Answer:
Longest chord of the circle = bore = 2 ten radius = 2 x 2.ix = 5.8 cm
(D) five.8 cm

vi. Radius of a circle with heart O is four cm. If fifty(OP) = iv.2 cm, say where bespeak P will prevarication ____.
(A) on the centre
(B) inside the circumvolve
(C) outside the circumvolve
(D) on the circle
Reply:
l(OP) > radius
∴Bespeak P lies in the outside of the circle.
(C) outside the circle

vii. The lengths of parallel chords which are on opposite sides of the eye of a circumvolve are 6 cm and viii cm. If radius of the circumvolve is 5 cm, so the distance between these chords is _____.
(A) two cm
(B) one cm
(C) eight cm
(D) 7 cm
Answer:

PQ = 8 cm, MN = 6 cm
∴ AQ = four cm, BN = 3 cm
∴ OQ² = OA² + AQ²
∴ five² = OA² + 4²
∴ OA^two = 25 – 16 = ix
∴ OA = iii cm
Besides, ON² = OB² + BN²
∴ 5² = OBⁱⁱ + iii²
∴ OB = 4 cm
At present, AB = OA + OB = 3 + 4 = 7 cm

Question ii.
Construct incircle and circumcircle of an equilateral ADSP with side 7.5 cm. Measure out the radii of both the circles and find the ratio of radius of circumcircle to the radius of incircle.
Solution:


Steps of structure:
i. Construct ∆DPS of the given measurement.
ii. Draw the perpendicular bisectors of side DP and side PS of the triangle.
three. Name the indicate of intersection of the perpendicular bisectors equally signal C.
4. With C as centre and CM as radius, describe a circle which touches all the three sides of the triangle.
v. With C as centre and CP every bit radius, draw a circle which passes through the 3 vertices of the triangle.

Radius of incircle = ii.two cm and Radius of circumcircle = iv.4 cm

Question 3.
Construct ∆NTS where NT = v.vii cm. TS = 7.5 cm and ∠NTS = 110° and describe incircle and circumcircle of it.
Solution:


Steps of construction:
For incircle:
i. Construct ∆NTS of the given measurement.
two. Draw the bisectors of ∠T and ∠S. Let these bisectors intersect at point I.
iii. Depict a perpendicular IM on side TS. Point M is the pes of the perpendicular.
iv. With I as centre and IM as radius, depict a circle which touches all the three sides of the triangle.
For circumcircle:
i. Draw the perpendicular bisectors of side NT and side TS of the triangle.
ii. Name the point of intersection of the perpendicular bisectors every bit betoken C.
3. Bring together seg CN
4. With C as centre and CN equally radius, draw a circle which passes through the three vertices of the triangle.

Question four.
In the adjoining figure, C is the centre of the circumvolve, seg QT is a bore, CT = 13, CP = 5. Find the length of chord RS.

Given: In a circle with centre C, QT is a diameter, CT = xiii units, CP = five units
To find: Length of chord RS
Construction: Join points R and C.
Solution:

i. CR = CT= 13 units …..(i) [Radii of the aforementioned circle]
In ∆CPR, ∠CPR = 90°
∴ CR² = CP^two + RP^two [Pythagoras theorem]
∴ 13² = 5^two + RP² [From (i)]
∴ 169 = 25 + RP^two [From (i)]
∴ RPⁱⁱ = 169 – 25 = 144
∴ RP = \(\sqrt { 144 }\) [Taking foursquare root on both sides]
∴ RP = 12 cm ….(2)

two. Now, seg CP _L chord RS [Given]
∴ RP = \(\frac { ane }{ two }\) RS [Perpendicular drawn from the centre of the circle to the chord bisects the chord.]
∴ 12 = \(\frac { ane }{ 2 }\) RS [From (ii)]
∴ RS = 2 x 12 = 24
∴ The length of chord RS is 24 units.

Question 5.
In the adjoining figure, P is the centre of the circle. Chord AB and chord CD intersect on the diameter at the indicate Eastward. If ∠AEP ≅ ∠DEP, then show that AB = CD.

Given: P is the centre of the circle.
Chord AB and chord CD intersect on the bore at the indicate E. ∠AEP ≅ ∠DEP
To show: AB = CD
Construction: Draw seg PM ⊥ chord AB, A-1000-B
seg PN ⊥ chord CD, C-N-D
Proof:

∠AEP ≅ ∠DEP [Given]
∴ Seg ES is the bisector of ∠AED.
Point P is on the bisector of ∠AED.
∴ PM = PN [Every point on the bisector of an angle is equidistant from the sides of the angle.]
∴ chord AB ≅ chord CD [Chords which are equidistant from the eye are congruent.]
∴ AB = CD [Length of congruent segments]

Question 6.
In the adjoining figure, CD is a diameter of the circle with middle O. Diameter CD is perpendicular to chord AB at point Eastward. Evidence that ∆ABC is an isosceles triangle.

Given: O is the centre of the circumvolve.
bore CD ⊥ chord AB, A-East-B
To evidence: ∆ABC is an isosceles triangle.
Proof:
diameter CD ⊥ chord AB [Given]
∴ seg OE ⊥ chord AB [C-O-Due east, O-E-D]
∴ seg AE ≅ seg BE ……(i) [Perpendicular drawn from the centre of the circumvolve to the chord bisects the chord]
In ∆CEA and ∆CEB,
∠CEA ≅ ∠CEB [Each is of 90°]
seg AE ≅ seg BE [From (i)]
seg CE ≅ seg CE [Mutual side]
∴ ∆CEA ≅ ∆CEB [SAS exam]
∴ seg AC ≅ seg BC [c. s. c. t.]
∴ ∆ABC is an isosceles triangle.

Maharashtra Board Class 9 Maths Chapter half dozen Circle Problem Set six Intext Questions and Activities

Question 1.
Every student in the grouping should do this action. Draw a circle in your notebook. Draw any chord of that circle. Describe perpendicular to the chord through the centre of the circumvolve. Mensurate the lengths of the two parts of the chord. Grouping leader should ready a table as shown below and inquire other students to write their observations in it. Write the property which yous have observed. (Textbook pg. no. 77)


Answer:
On completing the higher up tabular array, you volition detect that the perpendicular fatigued from the centre of a circle on its chord bisects the chord.

Question two.
Every educatee from the group should do this activity. Depict a circumvolve in your notebook. Draw a chord of the circle. Bring together the midpoint of the chord and eye of the circle. Measure the angles made by the segment with the chord.
Discuss nearly the measures of the angles with your friends. Which property do the observations suggest ? (Textbook pg. no. 77)

Answer:
The meausure of the angles fabricated by the drawn segment with the chord is 90°. Thus, we tin can conclude that, the segment joining the eye of a circumvolve and the midpoint of its chord is perpendicular to the chord.

Question 3.
Describe circles of user-friendly radii. Draw two equal chords in each circle. Draw perpendicular to each chord from the heart. Mensurate the distance of each chord from the eye. What exercise you notice? (Textbook pg. no. 79)
Answer:
Congruent chords of a circle are equidistant from the centre.

Question 4.
Mensurate the lengths of the perpendiculars on chords in the following figures.

Did y'all find OL = OM in fig (i), PN = PT in fig (two) and MA = MB in fig (iii)?
Write the holding which you have noticed from this action. (Textbook pg. no. 80)
Answer:
In each effigy, the chords are equidistant from the centre. Also, nosotros tin see that the measures of the chords in each circle are equal.
Thus, we tin conclude that chords of a circle equidistant from the eye of a circle are congruent.

Question 5.
Depict dissimilar triangles of unlike measures and draw in circles and circumcircles of them. Complete the tabular array of observations and discuss. (Textbook pg. no. 85)
Answer:

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