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Geometry and Mensuration: Test 25

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Question 1
R and S are two points on the sides PT and QT respectively of ΔPQR such that RS= 18 cm, TS= 5 cm and ÐRST= 90o. If tan ÐPQR= 3.6, then PT: TR =
A
QT: 2 TS
B
2 TS: QT
C
2 QT: TS
D
TS: 2 QT
Question 1 Explanation:
51
$ \displaystyle \begin{array}{l}\angle \,RST={{90}^{o}}\\RS=18\,cm\\TS=5\,cm\\\therefore \,\,\tan \,T=\frac{RS}{TS}=\frac{18}{5}=3.6\\\tan \,\angle PQR=3.6\\\therefore \angle T=\angle Q\\\therefore PT=PQ\\\angle T+\angle R={{90}^{o}}\\\Rightarrow 2\,\angle T+2\,\angle R={{180}^{o}}\\\angle T+\angle P+\angle Q={{180}^{o}}\\\Rightarrow \,2T+\angle P={{180}^{o}}\\\therefore \,\,\angle P=2\,\angle R\\\therefore \,\frac{PT}{TQ}=\frac{2TR}{TS}\end{array}$
Question 2
BL and CM are medians of ΔABC right-angled at A and BC= 5cm. If BL = 8cm, then the length of CM is
A
√61/2cm
B
√48/2cm
C
10√2cm
D
4 cm
Question 2 Explanation:
52
image-8
Question 3
ABCD is a rectangle where the ratio of the lengths of AB and BC is 3: 2. If P is the mid-point of AB, then the value of sin ∠CPB is
A
3/5
B
2/5
C
3/4
D
4/5
Question 3 Explanation:
53
image-9
Question 4
PQRS is a square, T is the mid-point of PQ and M is the mid-point of QR. PM and ST are joined and they meet at L. Then which of the following is correct?
A
TP: TL= 1: 2
B
PM = TS
C
∠PMS = ∠PMQ
D
∠PTS = ∠QPM
Question 4 Explanation:
54
$ \displaystyle \begin{array}{l}If\,\,\,PQ=2x,\,\,then\,\,QM=x\\\therefore \,\,PM=\sqrt{4{{x}^{2}}+{{x}^{2}}}=\sqrt{5}x\\Similarly,\\TS=\sqrt{4{{x}^{2}}+{{x}^{2}}}=\sqrt{5}x\end{array}$
Hence PM=TS
Question 5
If the sums of the interior angles of a regular polygon be 1080o, the number of sides of the polygon is
A
6
B
8
C
10
D
12
Question 5 Explanation:
Let the number of sides be x.
180(x-2)= 1080,
=> x - 2 = 6=> x = 8
Thus the correct option is (b)
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