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  • These tests focus on geometry and mensuration and are meant to indicate your preparation level for the subject.
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Geometry and Mensuration: Test 17

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Question 1
In a triangle ABC, the lengths of the sides AB, AC and BC are 3, 5 and 6 cm respectively. If a point D on BC is drawn such that the line AD bisects the ∠A internally, then what is the length of BD?
A
2 cm
B
2.25 cm
C
2.5 cm
D
3 cm
Question 1 Explanation: 
20
geometry-and-mensuration-test-17-question-1-pic-1
Question 2
If P and Q are the mid-points of the sides CA and CD respectively of a triangle ABC right angled at C. Then, the value of 4 (AQ2 + BP2) is equal to
A
4 BC2
B
5 AB2
C
2AC2
D
2BC2
Question 2 Explanation: 
21
$ \displaystyle \begin{array}{l}A{{D}^{2}}=\text{ }A{{C}^{2}}+\text{ }C{{D}^{2}}\\In\,\,PCB,\\\begin{array}{*{35}{l}} B{{P}^{2}}=\text{ }P{{C}^{2}}+\text{ }C{{B}^{2}} \\ A{{D}^{2}}+B{{P}^{2}}=\text{ }A{{C}^{2}}+C{{D}^{2}}+P{{C}^{2}}+\text{ }C{{B}^{2}} \\ =\text{ }\left( A{{C}^{2}}+C{{B}^{2}} \right)\text{ }+\text{ }\left( C{{D}^{2}}+\text{ }P{{C}^{2}} \right) \\ =\text{ }A{{B}^{2}}=\text{ }P{{D}^{2}} \\ \end{array}\\=A{{B}^{2}}{{\left( \frac{1}{2}\,AB \right)}^{2}}\\A{{D}^{2}}+\text{ }B{{P}^{2}}=\frac{4A{{B}^{2}}+A{{B}^{2}}}{4}\\Therefore\text{ }4\text{ }\left( A{{D}^{2}}+\text{ }B{{P}^{2}} \right)\text{ }=\text{ }5A{{B}^{2}}\end{array}$
Question 3
If the sides of a right triangle are x,x+1 and x-1,then the hypotenuse is
A
5
B
4
C
1
D
0
Question 3 Explanation: 
Since the triangle is a right angled triangle, We have, (x+1)2 = x2+(x-1)2
=> x2 +2x +1 = x2 + x2-2x +1
=> x2- 4x =0
=> x(x-4) =0
=>x = 4 [x cannot be equal to 0 ]
Therefore Hypotenuse = 4+1 = 5
Question 4
Let ABC be an acute-angled triangle and CD be the altitude through C. If AB = 8 and CD= 6, then the distance between the mid-points of AD and BC is
A
30
B
25
C
27
D
5
Question 4 Explanation: 
geometry-and-mensuration-test-17-question-4-pic-1
Let P and Q be the mid points of AD and BC respectively.
Draw QR perpendicular on AB. Since Q is the midpoint of BC, so R is the midpoint of BD. Hence by mid point theorem QR = 3.
Also AD = 4 = DB. So DR = PD = 2 Þ PR = 4. Hence by Pythagoras theorem, we have
PQ2 = PR2 + QR2 = 16 + 9 = 25
=>PQ = 5.
Question 5
In ΔABC, ∠B is a right angle, AC =6 cm, and D is the mid-point of AC. The length of BD is: 23
A
4 cm
B
√6 cm
C
3 cm
D
3.5 cm
Question 5 Explanation: 
AD = CD= 6/2 = 3 cm.
BD2=ADxCD = 9.
BD= 3 cm
. Correct option is (c)
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