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Geometry and Mensuration: Test 18

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Question 1
An angle is equal to 1/3rd of its supplement. Find its measure.
A
60o
B
80o
C
90o
D
45o
Question 1 Explanation: 
$\displaystyle \begin{array}{l}\begin{array}{*{35}{l}} Let\text{ }that\text{ }angle\text{ }be\text{ }p \\ Supplement\text{ }of\text{ }this\text{ }angle\text{ }=\text{ }180-p \\ \end{array}\\Therefore\,\,\,p=\frac{1}{3}\,\left( 180-p \right)\\\Rightarrow 3p=180-p\\\Rightarrow p={{45}^{o}}\end{array}$
Question 2
In a triangle ABC, ∠A = 90o and D is mid-point of AC. The value of BC2 – BD2 is equal to
A
AD2
B
2AD2
C
3AD2
D
4AD2
Question 2 Explanation: 
BC2 = AB2 + AC2
BD2 = AB2 + AD2
BC2 – BD2 = AB2 + AC2 – AB2 –AD2
= AC2 – AD2
= (AC– AD) (AC + AD)
=(2AD – AD) (2AD + AD)
= AD × 3AD = 3AD2
Question 3
The in-radius of an equilateral triangle is of length 3 cm. Then the length of each of its medians is
A
12 cm
B
9/2 cm
C
4 cm
D
9 cm
Question 3 Explanation: 
25
In equilateral triangle centroid, incentre,
orthocenter coincide at the same points
Height/3 = inradius
Therefore Height= Median= 3×3= 9 cm.
Question 4
If ABC is an equilateral triangle and D is a point on BC such that AD⊥BC ,then
A
AB: BD= 1: 1
B
AB: BD= 1: 2
C
AB: BD= 2: 1
D
AB: BD= 3: 2
Question 4 Explanation: 
26
Let AB = 2 unit.
As the triangle is an equilateral triangle, the perpendicular from A will bisect BC. (in this case at D) BD = 1 unit.
Required ratio AB: BD= 2: 1.
Correct option is (c)
Question 5
A ladder leans against a vertical wall. The top of the ladder is 8m above the ground. When the bottom of the ladder is moved 2 m farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder?
A
10 m
B
15 m
C
20 m
D
17 m
Question 5 Explanation: 
geometry-and-mensuration-test-18-question-5-pic-1-1
Let AB be the wall and AC is the ladder with C as the foot of the ladder.
Let BC = x and CD = 2. So the length of the ladder is x + 2 i.e. AC = x + 2.
Now in ΔABC, we have (x + 2)2 = x2 + 64
=> x2 + 4 + 4x = x2 + 64
=> 4x = 60 => x = 15m.
Hence the length of the ladder = 15 + 2 = 17m.
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