Tag: constructions of triangles

Questions Related to constructions of triangles

Construct a triangle $PQR$, whose perimeter is $22 cm$ and whose sides are in the ratio $2 : 4 : 5$. Measure the sides of the triangle.

maths construction of triangles constructions of triangles construction of triangles - ii constructions

  1. $5, 7, 10$

  2. $4, 8, 10$

  3. $5, 8, 9$

  4. None of these

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Correct Option: B
Explanation:

Let the sides be $2x,4x$ and $5x$

Perimeter $=22cm$
$2x+4x+5x=22$
$11x=22$
$x=2$
So the sides are 
$2x=2\times2=4cm$
$4x=4\times 2=8cm$
$5x=5\times 2=10cm$
Option $B$ is correct.

Construct a $\triangle ABC$ in which $AB= 5.4\ cm, \angle CAB= 45^{\circ}$ and $AC + BC= 9\ cm.$Then, $m\angle ACB$ is:

maths construction of triangles constructions of triangles construction of triangles - ii constructions

  1. $55^o$

  2. $75^o$

  3. $85^o$

  4. None of these

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Correct Option: B

For constructing a triangle whose perimeter and both base angles are given, the base length is equal to:

maths construction of triangles constructions of triangles construction of triangles - ii constructions

  1. the length of the perimeter

  2. the length of the largest side

  3. the difference between the largest and the shortest side

  4. None of these

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Correct Option: A
Explanation:

To construct a triangle when the perimeter and both base angles are given, we first draw the base with length equal to the perimeter of the triangle. After that we draw the complete triangle with proper method.

Choose the correct statement:

maths construction of triangles constructions of triangles construction of triangles - ii constructions

  1. Of all the line segments that can be drawn from a point outside a line, the perpendicular is the shortest.

  2. The difference of two sides of a triangle is equal to the third side.

  3. The sum of the three sides of a triangle is less than the sum of its three medians.

  4. If two sides of a triangle are unequal then the larger side has the smaller angle opposite to it.

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Correct Option: A
Explanation:

XY is a fixed line.

$LX=MO=NT=FP=a$
$In\triangle PXL$
we can see that
${ PX }^{ 2 }={ PL }^{ 2 }+{ LX }^{ 2 }$
$\implies\quad { PX }^{ 2 }={ PL }^{ 2 }+{ a }^{ 2 }\quad $-(1)
$ In\quad \triangle PMO:$
$ { PO }^{ 2 }+{ OM }^{ 2 }={ PM }^{ 2 }$
$ \implies\quad { PM }^{ 2 }={ PO }^{ 2 }+{ a }^{ 2 }\quad$ -(2)
$ In\quad \triangle PNT:$
$ { PT }^{ 2 }+{ TN }^{ 2 }={ PN }^{ 2 }$
$\implies\quad { PN }^{ 2 }={ PT }^{ 2 }+{ a }^{ 2 }\quad$ -(3)
$ In\quad \triangle PFN:$
$ { PN }^{ 2 }={ PF }^{ 2 }+{ FN }^{ 2 }$
$\implies\quad { PF }^{ 2 }={ PN }^{ 2 }-{ FN }^{ 2 }$
$ \because In\quad \triangle PFN,PN\quad is\quad $hypotenuse,
$ \therefore PN>PF\quad -(4)$
 comparing$(1)(2)(3)& (4)$
$ PL>PO>PT$
$\implies\quad PX>PM>PN$
$comparing \quad with(4)$
 $PN>PF$
$\implies\quad PX>PM>PN>PF$
Similarly,we can prove for right side of $PF.$
$\therefore PF$ is the shortest distance on $XY$ from $P.$
$\therefore A)$is correct.

For constructing a triangle when the base, one base angle and the difference between lengths of other two sides are given, the base length is equals to:

maths construction of triangles constructions of triangles construction of triangles - ii constructions

  1. The difference between lengths of other two sides

  2. The given base length

  3. The largest side

  4. None of these

Show answer
Correct Option: B
Explanation:

The length of the base is exactly given. Hence there is no need to extend the same.

The construction of a $\Delta ABC$ in which $BC=6$ $cm$ and $\angle B=50^\circ$, is not possible when $(AB-AC)$ is equal to:

maths construction of triangles constructions of triangles construction of triangles - ii constructions

  1. $5.6\ cm$

  2. $5\ cm$

  3. $6\ cm$

  4. $4.8\ cm$

Show answer
Correct Option: C
Explanation:

For construction of triangle sum of any two sides must be greater then third side

$\Rightarrow AC+BC>AB$
$AC+6>AB$
$6>AB-AC$
$AB-AC<6$
So $AB-AC\neg 6$
Option $C$ is correct.

The construction of $\Delta EFG$ when $FG=3$ $cm$ and m$\angle G=60^\circ$ is possible when difference of $EF$ and $EG$ is equal to:

maths construction of triangles constructions of triangles construction of triangles - ii constructions

  1. $3.2$ $cm$

  2. $3.1$ $cm$

  3. $3$ $cm$

  4. $2.8$ $cm$

Show answer
Correct Option: D
Explanation:
The difference possible isn't a constant

It could range from $0$ to less than $3\ cm$

It says less than $3\ cm$, and not equal to $3\ cm$ . This is so because sum of 2 sides is always greater than the third side in a triangle.

$\therefore EF-EG < 3 \ cm$

From the given options, D is the only possible solution as the rest are $\geq 3 \ cm$

For constructing a triangle whose perimeter and both base angles are given, the first step is to:

maths construction of triangles constructions of triangles construction of triangles - ii constructions

  1. Draw a base of any length

  2. Draw the base of length $=$ perimeter

  3. Draw the base angles from a random line.

  4. Draw a base of length $=\dfrac13 \times$ perimeter.

Show answer
Correct Option: B
Explanation:
Below are the steps for constructing a triangle whose perimeter and base angles are given.

Step 1 : Draw a line segment/base equal to perimeter 

Step 2 : From any point X draw ray at one of the given base angles. From any point Y draw ray at second of the given base angles

Step 3 : draw angle bisector of X and Y, two angle bisectors intersect each other at point A

Step 4 : Draw line bisector of XA and AY respectively these two line bisectors intersect XY at point B and C

Step 5 : Join A to B and A to C 

Step 6 : Triangle ABC is the required triangle.


Hence, the answer is 'Draw the base of length $=$ perimeter'.

The construction of $\triangle ABC$ in which $AB = 6\ cm, \angle A = 30^\circ$, is not possible when $AC+BC = $

maths construction of triangles constructions of triangles construction of triangles - ii constructions

  1. $6.3\ cm$

  2. $7.2\ cm$

  3. $5.6\ cm$

  4. $6.9\ cm$

Show answer
Correct Option: C
Explanation:

For constructing a triangle the sum of any two sides must be greater then third side.

Hence one side is $AB=6cm$
$\Rightarrow AC+BC>6cm$
So when $AC+BC=5.6cm$ it is not possible to construct a triangle.

The construction of $\triangle ABC$ in which $AB = 5\ cm, \angle A = 45^\circ$, is possible when $AC+BC = $

maths construction of triangles constructions of triangles construction of triangles - ii constructions

  1. $4.8\ cm$

  2. $5.6\ cm$

  3. $3.2\ cm$

  4. $2.8\ cm$

Show answer
Correct Option: B
Explanation:

For constructing a triangle the sum of any two sides must be greater then third side.

Hence one side is $AB=5cm$
$\Rightarrow AC+BC>5cm$
Here we have $AC+BC=5.6cm$ in option $C$
So it is possible to construct a triangle.