Practical Geometry Test

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Practical Geometry Test - 3

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Question 1
1 / -0

To construct a quadrilateral uniquely, it is necessary to have the knowledge of a least ___________ independent elements.

A
Four

B
Five

C
Three

D
Six

Solution

If the quadrilateral is irregular i.e it is not a rhombus or a parallelogram, then we need to know the lengths of all four sides plus one diagonal (i.e. at least five parts).
Example: For a quadrilateral $$ABCD,$$ if we know $$AB=a, BC=b, CD=c, DA=d$$ and $$AC=k$$ then we can construct the unique parallelogram subjected that
$$a+b>k$$ and $$c+d>k$$.
In fact we would construct two triangles $$\triangle ABC$$ and $$\triangle ADC$$ on common base $$AC$$, apexes $$B$$ and $$D$$ being on either side of $$AC,$$ using SSS method.
Question 2
1 / -0

Which of the following statement is/are correct?
We can construct a quadrilateral:

A
when its opposite sides are given

B
when its four angles and a side are given

C
when its diagonals are given

D
when its three angles and any two adjacent sides are given

Solution

To construct a unique quadrilateral, we will need a minimum of $$5$$ dimensions.
Here in option A, only four dimensions are provided, so a unique quadrilateral not possible because we don't know its angles.
In option B, we have five dimensions, but it does not result in a unique quadrilateral. we needed one more side length to construct uniquely.
In option C, It is not possible to construct a unique quadrilateral from only two diagonals given, unless it is a rhombus or square.
In option D, we have five dimensions. Here if we draw a side first then mark the angle on both ends then we can construct a quadrilateral uniquely.

Hence option D is correct.
Question 3
1 / -0

Is it possible to construct a rhombus with its diagonals equal to its side?

A
No

B
Sometimes

C
Always

D
Can't say

Solution

No, it is not possible to construct a rhombus with diagonals equal to its sides.
It can be constructed only if one of the diagonals is equal. However, if both diagonals are equal, it's not possible.
We can assume two equilateral triangles on both sides of line segment $$AC$$ with common base as $$AC$$.
Here, $$AB=AC$$, this implies that side $$AB$$ is equal to diagonal $$AC$$, but if you look at another diagonal $$BD$$, it does not come out to be same.
Hence, its not possible to have both the diagonals equal to its sides.
Question 4
1 / -0

We can construct a rhombus if its ___________ are given.

A
side

B
diagonals

C
angle

D
None of these

Solution

If we know the side only, we can't construct a unique rhombus because specific angles will be required to construct it, or if all angles are given, then also we can't construct a unique rhombus because a side length will be required to construct it.
But if the diagonals of rhombus are given we can construct a complete uniques rhombus.
Here are the steps:
1. Draw a diagonal $$AC$$.
2. Draw a perpendicular bisector of $$AC$$, let the perpendicular bisector and $$AC$$ intersects at $$O$$ ($$\because$$ Diagonals of rhombus intersects at right angle)
3. Taking $$O$$ as centre and half of the length of another diagonal as radius, mark arc on the perpendicular bisector at both sides of $$AC$$. Name these intersection points as $$B$$ and $$D$$
4. Join $$AB, AD, BC, CD$$.
5. $$ABCD$$ is the required rhombus.
Question 5
1 / -0

Which type of quadrilateral is this.

A
Rhombus

B
Rectangle

C
Trapezium

D
Kite

Solution

$$AB=BC=CD=AD$$ and
$$AD||BC, AB||DC$$
All sides have equal length. Opposite sides are parallel, and opposite angles are equal, its a Rhombus.
Question 6
1 / -0

Construct a rectangle $$ABCD$$, when its sides are $$6\ \text{cm}$$ and $$7.2\ \text{cm}$$.
Then, relation between diagonals $$AC$$ and $$BD$$ is:

A
$$AC=BD$$

B
$$AC<BD$$

C
$$AC>BD$$

D
Cannot say

Solution

Steps:
1) Draw a straight line $$AB$$ of length $$7.2\ \text{cm}$$
2) Draw perpendicular lines at $$A$$ and $$B$$ using protractor.
3) Using compass cut arc at the perpendicular from $$A$$ and $$B$$ of lengths $$6\ \text{cm}$$
4) Join these cuts with a line $$CD$$ as shown in figure.
5) Now measure the lengths of $$AC$$ and $$BD$$.

We get that the length of $$AC$$ is approx. $$9.37\ \text{cm}$$ which is same as length of $$BD$$

Hence, $$AC = BD$$.
Question 7
1 / -0

Which of the following statements is true for a rhombus?

A
It has only two pair of equal sides.

B
Two of its angles are at right angles.

C
Its diagonals bisect each other at right angles.

D
It is always a square.

Solution

Rhombus is a flat shape with 4 equal straight sides.All sides have equal length.Opposite sides are parallel, and opposite angles are equal.The altitude is the distance at right angles to two sides.And the diagonals "p" and "q" of a rhombus bisect each other at right angles.
So (C) is correct.
Answer (C) Its diagonals bisect each other at right angles.
Question 8
1 / -0

Construct a quadrilateral $$ABNV$$ where $$AV = 7\ cm, BA = 8\ cm, BN = 10\ cm, NV = 5\ cm$$ and $$BV = 11\ cm.$$
The relation between diag. $$AN$$ and diag. $$BV$$ is

A
$$AN > BV$$

B
$$AN < BV$$

C
$$AN = BV$$

D
None of the above

Solution

1. Draw a line segment $$BV = 11\ cm$$.
2. With $$10\ cm$$ as radius, draw an arc from $$B$$.
3. Similarly, with radius as $$5$$ cm from $$V$$, cut the arc just drawn.
4. They will intersect at $$N$$. Join $$BN$$ and $$VN$$.
5. With $$7\ cm$$ as radius, draw an arc from $$V$$. Similarly, with radius as $$8\ cm$$ from $$B$$, cut the arc just drawn.
6. They will intersect at $$A$$. Join $$AB$$ and $$VA$$.
7. $$ABNV$$ is the required quadrilateral.
8. Upon measuring the distance between $$A$$ and $$N$$, we get $$AN =10\ cm $$
Therefore, $$BV>AN$$
Question 9
1 / -0

Construct a quadrilateral ABCD, when AB = 4 cm, BC = 5 cm, CD = 6.5 cm, $$\angle ABC\, = 105^{\circ}$$ and $$\angle DCB\, = 80^{\circ}$$. The length of AD is

A
$$2.9$$cm

B
$$3.5$$cm

C
$$5.5$$cm

D
$$4.5$$cm

Solution

1. Draw a line segment $$BC = 5\ cm$$.
2. Draw a line segment $$XB$$ making angle of $$105˚$$ at $$B$$
3. With $$4\ cm$$ as radius, draw an arc from $$B$$, name it as $$A$$.
4. Draw a line segment $$YC$$ making angle of $$80˚$$ using protractor at $$C$$.
5. With radius as $$6.5$$ cm from $$C$$, cut the arc just drawn, name it as $$A$$.
6. Join $$AD$$.
7. $$ABCD$$ is the required quadrilateral.
8. Measure the length of $$AD = 5.5\ cm$$.
Question 10
1 / -0

In a quadrilateral $$ABCD, AB=AC=10cm$$ and $$DB=DC=5cm$$. Its diagonals intersect each other at $$90^o$$.what type of quadrilateral is it.

A
Trapezium

B
Kite

C
Rhombus

D
Rectangle
Solution
In kite quadrilateral,
Two pairs of sides are of equal length.
One pair of diagonally opposite angles is equal.
The diagonals intersect at 90°.