Elitmus
Exam
Numerical Ability
Geometry
PR is a tangent to a circle at point P.Q is another point on the circle such that PQ is the diameter and RQ cuts the circle at point M. If the radius of the circle is 4 units and PR=6 units then find the ratio of the perimeter of triangle PMR to the triangle PQR
a)11/20
b)3/5
c)13/20
d)18/25
Read Solution (Total 10)
-
- The ans is (b) 3/5
Sol:
I hope the geometry of this questn is clr.
Nw since PR is the tangent,Hence angle QPR is 90.
=>Triangle PQR is rit-angled triangle.
=>hyp QR=10. (Pythagoras Theorem)........(1)
Also PM is perpendicular to QR (Since triangle PQR is made inside circle with diameter as hyp
hence PMQ is rit-angled...itz a property of circle)
=>three triangles PQR,PMQ & PMR are similar to each other.
(For similarity of such type of triangles u can refer any 10th std book).
NOw in that pic consider the triangles PQR & PMR.
PQR~PMR
hence PR/RM=QR/PR.
=>PR^2=RM*QR
putting the values & from(1)
=>RM=18/5....(2)
Also
PQR~PQM
=>PQ/QM=QR/PQ
=>PQ^2=QM*QR
putting the values & from(1)
=>QM=32/5.....(3)
Also,
PQM~PMR
=>PM/QM=RM/PM
=>PM^2=QM*RM
from(2) & (3)...
PM=24/5......(4)
Hence peimeter of triangle PMR=PM+RM+PR
Putting the obtained values perimeter=72/5
Also perimeter of triangle PQR=PQ+QR+PR=24.
Hence th ratio PMR/PQR=72/(5*24)=3/5
:-)) - 12 years agoHelpfull: Yes(86) No(9)
- mr ashu dont be oversmart
- 10 years agoHelpfull: Yes(14) No(3)
- b)3/5 is the ratio of the perimeter of triangle PMR to the triangle PQR.
- 12 years agoHelpfull: Yes(8) No(12)
- Answer
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 PMRPerimeter of triangle PQR=14.4/24=3/5
- 9 years agoHelpfull: Yes(7) No(0)
- explian the ans plz
- 12 years agoHelpfull: Yes(1) No(6)
- the ans is (c)
13/20 - 11 years agoHelpfull: Yes(0) No(7)
- 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
PMPR=PQRQPM6=810
PM = 4810 = 4.8 units ----(3)
tan PRM = tan PRQ
PMRM=PQPR4.8RM=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=14.424=144/240=3/5 - 9 years agoHelpfull: Yes(0) No(0)
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
PMPR=PQRQPM6=810
PM =
48
10
4810 = 4.8 units ----(3)
tan PRM = tan PRQ
PM
RM
=
PQ
PR
4.8
RM
=
8
6
PMRM=PQPR4.8RM=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
Perimeter of triangle PMRPerimeter of triangle PQR=14.424=144240=35
- 8 years agoHelpfull: Yes(0) No(0)
- PQ=8(diameter of the circle)
PR=6
so, QR=10 (pythagorean theorem)
angle QMP=90 degree (angle subtanding diameter of a circle)
angle QMP=angle RMP=90 degree
assume QM=a and MP=b
MR=10-a
we get equations, a^2 + b^2=8^2......(i)
and, b^2 + (10-a)^2=6^2......(ii)
solving these two equations, we get, a=6.4 and b=4.8
so, perimeter(triangle PMR)/ perimeter(triangle PQR)=(b+10-a+6)/(8+10+6)=14.4/20=18/25......option (d) - 7 years agoHelpfull: Yes(0) No(0)
- Ans:13/20
PQ-8
PR-6
QR-10 by pythagores theorm
let PM be k
PQR right 3gle
PM- x
MR- 10-x
PMR also right 3gle as formed by diameter
64=k^2+x^2
36=(10-x)^2+k^2
solving for x gives
x=(64/10)
then k=(48/10)
perimeter of PMR=6+(48/10)+(48/10)
perimetr of PQR =6+8+10
ratio
13/20 - 5 years agoHelpfull: Yes(0) No(3)
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