Latest Elitmus Aptitude Question SOLUTION: PR is tangent to a circle at point P. Q is another point on circle such that PQ is diameter of circle and RQ cuts circle at M. if radius of circle is 4 units and PR=6units. find ra

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(#M40153809) ELITMUS Numerical Ability question geometry Keep an EYE

PR is tangent to a circle at point P. Q is another point on circle such that PQ is diameter of circle and RQ cuts circle at M. if radius of circle is 4 units and PR=6units. find ratio of triangle PMR to PQR. optn
a)11/20
b)3/5
c)13/20
d)18/25

Asked In Elitmus sukriti singh (8 years ago)

Unsolved Read Solution (15)

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Rohit   (8 years ago)

PR^2+PQ^2=QR^2
8^2+6^2=QR^2
QR=10

NOW TO FIND THE RATIO OF PERIMETER OF TWO TRIANGLE TAKE THE OF UNCOMMON ANGLE i.e,
angle QPR

so COS (QPR)= COS (PR/QR)
=6/10
=3/5 (ANSWER)

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pratik raj  (8 years ago)

let PM = y
QM= x
MR = 10-x
In triangle PQM,
PM^2+QM^2 = PQ^2
=> x^2+y^2=64 .....(1)
In triangle PMR,
PM^2+ MR^2= PR^2
y^2+(10-x)^2=36......(2)
solving for x and y
x=6.4
y=4.8
taking permineter of triangle PQR =24
perimeter of triangle PMR = 14.4
take ratio 14.4/24= 0.6
so option (c) is right..

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anupriya verma  (8 years ago)

PQ^2+PR^2=QR^2
QR=10

PM^2+RM^2=6^2(PR=6)
PM^2+QM^2(which is (10-RM)^2)=8^2(pq=8)
using above to equations we found rm=3.6 (so qm=10-3.6=4.8)and pm=4.8
now finding perimeter of both triangles we get ratio as 3/5

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SANTU MAITY  (8 years ago)

Given PQ = 8 cm (Diameter of Circle)
PR = 6 cm (Tangent to the circle)
Therefore QR = 10cm
Let QM = x and PM = y ; MR = 10-x
PM^2 + MQ^2 = PQ^2 or x^2 + y^2 = 64 ........ (i)

MR^2 + PM^2 = PR^2

or (10 - x)^2 + y^2 = 36 ...... (ii)
Subtracting (ii) from (i)


(x + 10 - x )(x - 10 +x) = 28

10(2x -10) = 28

x = 6.4
By solving we get y=4.8
(perimeter of PMR)/(perimeter of PQR)=(4.8+3.6+6)/(8+10+6)=14.4/24=3/5
answer option (b)3/5

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Sourav  (8 years ago)

U ll get QR=10 , PMR and PQR r two right ang. triangles

(PR common side, PRQ common angle , two right angles RPQ and PMR ) So they are similar.

now as per the similarity rule (ratio of sides = ratio of triangles)
i.e U can take PR/RQ=RPM/PQR = 6/10, or find MR and compare u ll get same ans.

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Anjali Sadhukhan  (8 years ago)

" RQ cuts circle at M"..@rohit ,justify this first in your answer.then i ll hit a like on it :)

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sukriti singh  (8 years ago)

perimeter

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Mithilesh Times  (8 years ago)

Ans. 3/5
Given PQ = 8 cm (Diameter of Circle)
PR = 6 cm (Tangent to the circle)
Therefore QR = 10cm
Let QM = x and PM = y ; MR = 10-x
PM^2 + MQ^2 = PQ^2 or x^2 + y^2 = 64 ........ (i)

MR^2 + PM^2 = PR^2

or (10 - x)^2 + y^2 = 36 ...... (ii)
Subtracting (ii) from (i)


(x + 10 - x )(x - 10 +x) = 28

10(2x -10) = 28

x = 6.4
By solving we get

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James Anderson  (7 years ago)

Anjali can you please shut your mouth

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sangeeta kumari  (6 years ago)

@Jignesh ,can u talk politely to Anjali!such a barbarian u r.learn manners

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shwetha  (6 years ago)

Since PR is tangent, angle RPQ = 90°. Hence QPR is a right angled triangle.

Given that PR = 6 units ----(1)
PQ = Diameter of the circle = 2×4 = 8 units ----(2)

Using Pythagorean theorem, QR2 = PR2 + PQ2 = 62+82 = 100
=> QR = 10 units ----(3)

Angle inscribed in a semicircle is always 90° (Thales' theorem)
=> angle PMQ is 90°

sin PRM = sin PRQ
PM/PR=PQ/RQ
PM/6=8/10
=> PM = 48/10 = 4.8 units ----(3)

tan PRM = tan PRQ
PM/RM=PQ/PR
4.8/RM=86
=> RM = 3.6 units ----(4)

Perimeter of triangle PMR = PM + RM + PR = 4.8 + 3.6 + 6 = 14.4 units
Perimeter of triangle PQR = PQ + QR + PR = 8 + 10 + 6 = 24 units

Perimeter of triangle PMR/Perimeter of triangle PQR=14.4/24=144/240= 3/5

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shishir srivastava  (8 years ago)

ratio of triangle....i mean ratio of what..their areas or their perimeters

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Anjali Sadhukhan  (8 years ago)

isn't the answer comes 13/20 guys!!!!

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kumar mohit  (8 years ago)

anjali can u post ur solution

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richa   (8 years ago)

rohit ....how ratio of perimeter is equal to cosin of angle ??? please explain...

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