Mathematics · Quantitative Aptitude

Geometry of Triangles and Angles

846 Questions

Triangle and angle geometry covers angle sums, properties of equilateral shapes, and right triangles. These principles form the basis of advanced quantitative aptitude sections. Practicing spatial problems helps secure points in exams.

Triangle angle sumsPythagorean theoremProperties of equilateral trianglesComplementary and supplementary anglesRight triangle formulas

Geometry of Triangles and Angles Questions

Multiple choice mathematics and statistics angle and its measurement directed angles

A half turn about O is a rotation through angel of ____ or ____

  1. $-90^0, +90^0$
  2. $+180^0, -180^0$
  3. $+360^0, -360^0$
  4. $-270^0, +270^0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A half turn is a rotation of 180 degrees. In either clockwise or counter-clockwise direction, this is represented as +180 or -180 degrees.

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

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

  1. $55^o$
  2. $75^o$
  3. $85^o$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a triangle construction where AB, angle A, and the sum AC + BC are given, the angle ACB is determined by the geometry of the construction. For these specific dimensions, the resulting angle is 75 degrees.

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

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

  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
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

Choose the correct statement:

  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.
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

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:

  1. The difference between lengths of other two sides

  2. The given base length

  3. The largest side

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

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

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

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:

  1. $5.6\ cm$
  2. $5\ cm$
  3. $6\ cm$
  4. $4.8\ cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

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:

  1. $3.2$ $cm$
  2. $3.1$ $cm$
  3. $3$ $cm$
  4. $2.8$ $cm$
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $\geq 3 \ cm$
Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

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

  1. $6.3\ cm$
  2. $7.2\ cm$
  3. $5.6\ cm$
  4. $6.9\ cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

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

  1. $4.8\ cm$
  2. $5.6\ cm$
  3. $3.2\ cm$
  4. $2.8\ cm$
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

Construct a $\triangle ABC$ in which:
$AB= 5.4\ cm$, $\angle CAB= 45^{0}$ and $AC\, +\, BC= 9\ cm$. Then the length of $AC$ (in $cm.$) is:

  1. $4$
  2. $7$
  3. $5$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the construction method for a triangle with given base, base angle, and sum of other two sides, the length of AC is derived from the geometric properties of the resulting triangle. For these values, AC is 5 cm.

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

The construction of $\Delta LMN$ when $MN=7$ $cm$ and $m\angle M=45^\circ$ is not possible when difference of $LM$ and $LN$ is equal to:

  1. $4.5$
  2. $5.5$
  3. $6.5$
  4. $7.5$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The triangle inequality rule states that the length of a side of a triangle is less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.


In $\triangle LMN$, if $MN=7 \ cm$ then $LM-LN<7 \ cm$

This is not possible, from the given options, if $LM-LN=7.5 \ cm$

Option D.

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

Which of the following could be the value of $AC-BC$ in the construction of a triangle $ABC$ in which base $AB = 5 cm, \angle A = 30^{\circ}$?

  1. $5.5$
  2. $5$
  3. $2.5$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The triangle inequality rule states that the length of a side of a triangle is less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.


In $\triangle ABC$, if $AB=5 \ cm$ then $AC-BC<5 \ cm$

This is not, from the given options, if $AC-BC=2.5 \ cm$

Option C

Multiple choice maths construction of triangles constructions of triangles construction of triangles - ii constructions

The construction of $\Delta LMN$ when $MN=6$ $cm$ and $m\angle M=45^\circ$ is not possible when difference between $LM$ and $LN$ is equal to:

  1. $6.9$ $cm$
  2. $5.2$ $cm$
  3. $5$ $cm$
  4. $4$ $cm$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a triangle sum of length of $2$ sides is $>$ third side.

or 

Difference of length of two sides is less than third side.

$LM,MN,NL$ are the length of sides.

$|LM-NL|<MN$

$|LM-NL|<6$

So construction of triangle is not possible as $|LM-NL|=6.9cm$

Multiple choice maths geometrical constructions construction of triangles constructions of triangles construction of triangles - ii

To construct a triangle similar to a given triangle ABC with its sides 6/5th of the corresponding sides of $\Delta$ABC. Correct order of steps of construction -
(a) Draw a ray AX inclined at certain angle with AB on opposite side of C.
(b) Starting from A, cut off six equal line segments AX$ _1$, X$ _1$X$ _2$, X$ _2$X$ _3$, X$ _3$X$ _4$, X$ _4$X$ _5$ and X$ _5$X$ _6$ on AX.
(c) Draw a line B'C' parallel to BC to intersect AC produced at C'
(d) Join X$ _5$B and draw a line X6B' parallel to X5B, to intersect AB produced at B'.

  1. abcd

  2. acbd

  3. abdc

  4. adcb
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To construct a triangle like the one given in the following question, the given steps are to be followed:

1. Draw a ray AX inclined at a certain angle with AB on opposite side of C.
2. Starting from A, cut off six equal line segments $AX _1, X _1X _2, X _2X _3, X _3X _4, X _4X _5 and X _5X _6$ on $AX$.
3. Join $X _5B$ and draw a line $X _6B'$ parallel to $X _5B$, to intersect $AB$ produced at $B'$.
4.Draw a line $B'C'$ parallel to $BC$ to intersect $AC$ produced at $C'$.

Multiple choice maths geometrical constructions construction of triangles constructions of triangles construction of triangles - ii

Construct a $\Delta ABC$, whose perimeter is $10.5  cm$ and base angles are $60^o$ and $45^o$. Find the third angle.

  1. $75^o$
  2. $45^o$
  3. $90^o$
  4. $60^o$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
  1. Draw a line segment PQ such that AB + BC + AC = PQ.
    2. Construct $\angle XPQ =\angle B = { 60 }^{ 0 }\ and\ \angle YQP =\angle C =45^{ 0 }$
    3. Bisect $\angle XPQ and \angle YQP$ and their bisectors will meet at A.
    4. Draw the perpendicular bisector DE of AP which meets PQ  at B. 
    5. Draw the perpendicular bisector FG of AQ which meets PQ  at C.
    6. Join AB & AC.
    7. ABC is the required triangle.
    Sum of interior angle of triangle= $180^o$
    so third angle = $180^o-45^o-60^o\ = 75^o $