# Free Constructions 01 Practice Test - 10th Grade

A line segment AB is divided in the ratio m:n where m and n are co-prime, using a single ray AX. The number of arcs to be drawn on AX is _____.

A.

m

B.

n

C.

m+n

D.

m-n

#### SOLUTION

Solution : C

The number of arcs to be drawn, when a single ray is used to divide a line segment in the ratio m:n, where m and n are co-prime, is m+n

Which similarity is used to prove that the constructed triangles are similar?

A.

SAS Similarity

B.

AA Similarity

C.

SSS Similarity

D.

ASA Similarity

#### SOLUTION

Solution : B

AA(Angle-Angle) similarity is used to prove that the constructed triangles are similar.

What will the ratio AB:AC be if C divides the line segment AB in the ratio 5:12?

A.

5:12

B.

17:12

C.

12:17

D.

17:5

#### SOLUTION

Solution : D Given ACBC=512

Therefore, BCAC=125

ABAC=AC+BCAC=1+BCAC=1+125=175

So, the required ratio =17:5

What is the ratio ACBC for the following construction:
A  line segment AB is drawn.
A single ray is extended from A and 12 arcs of equal lengths are cut, cutting the ray at A1,A2A12.
A line is drawn from A12 to B and a line parallel to A12B is drawn, passing through the point A6 and cutting AB at C.

A.

1:2

B.

1:1

C.

2:1

D.

3:1

#### SOLUTION

Solution : B In the construction process given, triangles AA12B and AA6C are similar.

Hence, we get ACAB=612=12.

By construction BCAB=612=12.

ACBC=ACABBCAB

=1212=1.

State whether true or false:
The line segment AB can be divided in a ratio only if the given ratio is less than 1.

A.

True

B.

False

#### SOLUTION

Solution : B

False. As long as the ratio is any positive rational number, a line segment can be divided in that ratio.

You are given a circle with radius 'r' and centre 'O'. You are asked to draw a pair of tangents which are inclined at an angle of 60° with each other, from a point E.
Refer to the figure and select the option which would lead you to the required construction. The distance d is the distance OE. A.

Using trigonometry, arrive at d = r and mark E.

B.

Construct the MNO as it is equilateral triangle.

C.

Mark M and N on the circle such that MOE = 60 and NOE = 60.

D.

Mark M and N on the circle such that MOE = 120 and NOE = 120.

#### SOLUTION

Solution : C

Since the angle between the tangents is 60°, we get MON=120
(As MONE is a quadrilateral and sum of angles of a quadrilateral is 360).
Hence, ΔMNO is NOT equilateral.

Since E is outside the circle, d can not be equal to r.

We know that MOE = 60°, following are the steps of construction:

1. Draw a ray from the centre O.

2. With O as centre, construct MOE = 60° .

3. Now extend OM and from M, draw a line perpendicular to OM. This intersects the ray at E. This is the point from where the tangents should be drawn and EM is one tangent.

4. Similarly, EN is another tangent.

The line segment AB was divided in the ratio 4:7 by taking 2 rays. The number of arcs to be made on the ray AX is

___

#### SOLUTION

Solution :

The line segment AB is divided in the ratio 4:7. The number of divisions to be made on the ray AX is 4 + 7 = 11.

Steps to divide a line segment  AB in the given ratio 3 : 2 by corresponding angles method is given. Choose the correct order.

1. Draw any ray AX making an acute angle with AB
2.Locate 5 pointsA1,A2,A3....A5 on ray AX
3. Join BA5
4.Draw a line parallel to BA5 through A3 to AB.

A. 2,1,3,4
B. 1,2,3,4
C. 1,3,2,4
D. 2,3,1,4

#### SOLUTION

Solution : B

To divide a line segment by corresponding angle method, we have to follow the steps
1. Draw any ray AX making an acute angle with AB
2.Locate (m+n) A1,A2,A3..Am+n points in AX
3. Join BAm+n
4.Draw a line parallel to BAm+n through Am to AB. In the given image, segment AB has been divided in the ratio 3:2. This is done by
1. Draw any ray AX making acute angle with AB.
2. Draw a ray BY parallel to AX by making ABY=BAX
3. Locate the points A1,A2,A3...A3 on AX and  B1,B2 on BY such that AA1=A1A2=BB1=B1B2
4. Join A3B2by using which of the properties of parallel lines? A. Corresponding angle are equal
B. Alternate interior angles are equal
C. Co-interior angle are supplementary
D. Perpendicular bisector theorem

#### SOLUTION

Solution : B Here we use alternate interior angles are equal then the lin4es are parallel and XAB=ABY as AX parallel to BY.

In the given image, segment AB has been divided in the ratio 3:2. This is done by
1) Drawing BAX
2) marking equal lengths AA1,A1A2,A2A3,A3A4 & A4A5
3) Point A5 is joint to point B
4) A3C is drawn parallel to A5B by using which of the properties of parallel lines? A. Corresponding angles are equal
B. Alternate interior angles are equal
C. Co-interior angles are supplementary
D. Perpendicular bisector theorem

#### SOLUTION

Solution : A Here we use the principle that when corresponding angles are equal, the lines are parallel.
AA3C=AA5C as A3C is parallel to A5B.