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Geometry and Mensuration: Test 26

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Question 1
The number of sides in two regular polygons are in the ratio 5: 4 and the difference between each interior angle of the polygons is 3o. Then the number of sides is
A
15, 12
B
30, 24
C
10, 8
D
21, 16
Question 1 Explanation: 
$\displaystyle \begin{array}{l}Let\text{ }the\text{ }number\text{ }of\text{ }sides\text{ }be\text{ }5xand\text{ }4xrespectively\\\frac{\left( 2\times 5x-4 \right){{90}^{o}}}{5x}-\frac{\left( 2\times 4x-4 \right)\times {{90}^{o}}}{4x}={{3}^{{}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ \begin{array}{l}each\,\,\,\,\operatorname{int}erior\,\,angle\,\\=\left( \frac{2n-4}{n} \right)\times {{90}^{o}}\end{array} \right]\\\Rightarrow \left( 10x-4 \right)\times {{360}^{o}}-\left( 8x-4 \right)\times {{450}^{o}}=20x\times {{3}^{o}}\\\Rightarrow 120x-48-120x+60=2x\\\Rightarrow 2x=12\Rightarrow x=6\\Therefore\,\,\,Number\,\,of\,\,\,sides\\=30\,\,and\,\,24\end{array}$
Question 2
ABCD is a cyclic trapezium whose sides AD and BC are parallel to each other. If ∠ABC= 72o, then the measure of the BCD is
A
162o
B
18o
C
108o
D
72o
Question 2 Explanation: 
55
Since ABCD is a cyclic trapezium, so its non parallel sides will be equal. Also AD || BC, so ∠ABC= ∠BCD => ∠BCD=72o Correct option is (d)
Question 3
If an exterior angle of a cyclic quadrilateral be 50o, then the interior opposite angle is:
A
130o
B
40o
C
50o
D
90o
Question 3 Explanation: 
The immediate internal angle = 1800-500 = 1300
The opposite internal angle = 1800 - 1300 = 500.
Correct option is (c).
Question 4
The ratio of the length of the parallel sides of a trapezium is 3: 2. The shortest distance between them is 15cm. If the area of the trapezium is 450 cm2, the sum of the lengths of the parallel sides is
A
15 cm
B
36 cm
C
42 cm
D
60 cm
Question 4 Explanation: 
$ \begin{array}{l}\frac{1}{2}\times 15\times (\text{Sum of the length of two parallel sides})=450\\\text{Sum of length of parallel sides = 60}\text{.}\end{array}$
Question 5
If the incentre of an equilateral triangle lies inside the triangle and its radius is 3 cm, then the side of the equilateral triangle is
A
9√3cm
B
6√3cm
C
3√3cm
D
6 cm
Question 5 Explanation: 
$ \begin{array}{l}Let\text{ }the\text{ }side\text{ }of\text{ }equilateral\text{ }triangle\text{ }be\text{ }x.\\\frac{1}{3}.\frac{\sqrt{3}}{2}x=3\\=>x=6\sqrt{3}\end{array}$
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