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Geometry and Mensuration: Test 28

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Question 1
ABCD is a rhombus. AB is produced to F and BA is produced to E such that AB= AE= BF. Also ED and CF are produced to meet at G. Then:
A
ED > CF
B
EG⊥GF
C
ED2 +CF2= EF2
D
ED||CF
Question 1 Explanation: 
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Here AB= AE= BF.......(1)Given ABCD is a rhombus.
Therefore, AB = BC = CD = AD ... (2)
On equating (1) and (2), we get
BC = BF
⇒∠4 = ∠3 [Angles opposite to equal sides are equal]
Again, ∠B is the exterior angle of triangle BFC.
Therefore, ∠2 = ∠3 + ∠4 = 2∠4 ... (3)
Similarly, AE = AD ⇒∠5 = ∠6
Also, ∠A is the exterior angle of triangle ADE
⇒∠1 = 2∠6 ... (4)
Also, ∠1 +∠2 = 180° [consecutive interior angles]
∴ 2∠6 + 2∠4 = 180°
⇒∠6 + ∠4 = 90° ... (5)
Now, in triangle EGF, by angle sum property of triangle
⇒ 90°+ ∠G = 180° [using (5)]
Thus, EG ⊥ FG.
Question 2
Two equal circles of radius 4 cm intersect each other such that each passes through the centre of the other. The length of the common chord is:
A
$ \displaystyle 2\sqrt{3}$
B
$ \displaystyle 4\sqrt{3}$
C
$ \displaystyle 2\sqrt{2}$
D
$ \displaystyle 8$
Question 2 Explanation: 
$ \begin{array}{l}The\text{ }length\text{ }of\text{ }the\text{ }common\text{ }chord\text{ }=\text{ }\\2\text{ }\times \text{ }length\text{ }of\text{ }the\text{ }altitude\text{ }of\text{ }the\text{ }equilateral\text{ }triangle\text{ }of\text{ }side\text{ }4\text{ }cm.\\Thus\text{ }the\text{ }length\text{ }of\text{ }common\text{ }chord\text{ }is\\2\times \frac{\sqrt{3}}{2}\times 4\\=4\sqrt{3}\end{array}$
Question 3
The length of two chords AB and AC of a circle are 8 cm and 6 cm and ∠BAC= 90o, then the radius of circle is
A
25 cm
B
20 cm
C
4 cm
D
5 cm
Question 3 Explanation: 
56
$ \begin{array}{l}The\text{ }diameter=\sqrt{{{8}^{2}}+{{6}^{2}}}=10\\Thus\text{ }the\text{ }radius\text{ }=5.\end{array}$
Question 4
The tangents are drawn at the extremities of a diameter AB of a circle with centre P. If a tangent to the circle at the point C intersects the other two tangents at Q and R, then the measure of the ∠QPR is
A
45o
B
60o
C
90o
D
180o
Question 4 Explanation: 
57
∠QCP= 900,
∠QAP=900.
AQ=QC.
Thus AQCP is a square.
∠QPC= 450
Similarly ∠RPC=450
Thus ∠QPR = 900.
Correct option is (c)
Question 5
The circumcentre of a triangle ΔABC is O. If ∠BAC= 85o and BCA= 75o, then the value of ∠OAC is
A
40o
B
60o
C
70o
D
90o
Question 5 Explanation: 
58
∠BAC=85, ∠BCA=75
∠AOC=360-2(BAC+BCA)
=360 – 320=40
Thus ∠OAC= ½ (180-40)=70
Correct option is (c)
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