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 maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

If $O$ is any point in the interior of $\Delta ABC$. then "$2(OA+OB+OC)=(AB+BC+CA)$" the statement  is?

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By the triangle inequality, for any point O inside a triangle, the sum of the distances from O to the vertices is less than the semi-perimeter, and specifically, the inequality 2(OA+OB+OC) < (AB+BC+CA) holds true.

Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

In $\triangle {ABC},ABC,APQ$ and $\overline { PQ } \parallel \overline { BC } $. If $PQ=5,AP=4,AB=12$, then $BC=$_____

  1. $9.6$
  2. $20$
  3. $15$
  4. $5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since PQ is parallel to BC, triangle APQ is similar to triangle ABC. Therefore, PQ/BC = AP/AB. Substituting the values: 5/BC = 4/12. This simplifies to 5/BC = 1/3, so BC = 15.

Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

If a,b,c are the sides of a triangle ABC, then $\sqrt{a} + \sqrt{b} - \sqrt{c} $  is always:

  1. negative

  2. Positive

  3. non - negative

  4. non - positive
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If $a,b$ and $c$ are the sides of the triangle then $\sqrt a  + \sqrt b  - \sqrt c $ it always positive because the sum of two sides of the triangle is always greater than the third side.

 

Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

In a $\Delta ABC$, side AB has the equation $2x+3y=29$ and the side AC has the equation, $x+2y=6$. If the mid-point of BC is (5, 6), then the equation of BC is

  1. $x-y=-1$
  2. $5x-2y=13$
  3. $21x+31y=291$
  4. $3x-4y=-9$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The intersection of AB and AC gives vertex A. Solving 2x+3y=29 and x+2y=6: x = 6-2y, so 2(6-2y)+3y=29, 12-4y+3y=29, -y=17, y=-17, x=40. The line BC passes through (5,6) and its slope can be found by relating it to the median or vertex properties, but checking the options, x-y=-1 passes through (5,6) since 5-6=-1.

Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

In a triangle ABC , if AB , BC and AC are the three sides of the triangle , then which of the following statements is necessarily true ?

  1. $\displaystyle AB + BC < AC$
  2. $\displaystyle AB + BC > AC$
  3. $\displaystyle AB + BC = AC$
  4. $\displaystyle AB^2 + BC^2 = AC^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
The sum of any two sides of a triangle is greater than the third side .

In $\triangle ABC, AB, BC$ and $AC$ are the three sides ,

Now ,

$AB + BC > AC$
Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

The sum of the three altitudes of a triangle is $......$ than its perimeter

  1. Less

  2. More

  3. Equal

  4. Less than or more than
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given: $\triangle ABC$, AD, BE and CF are perpendiculars from A, B and C respenctively on the corresponding sides.

Now, In $\triangle ADB$,
$AD < AB$ (Hypotenuse is the longest side)
Similarly in $\triangle ADC$,
$AD < AC$ (Hypotenuse is the longest side)
Hence, $2AD < AB  + AC$ (1)
Similarly we can say, $2BE < BC + AB$ (2)
and $2 CF < AC + BC$ (3)
or adding (1), (2), (3)
$2 (AC + AB + BC) > 2(AD + BE + CF)$
$AC + AB + BC > AD + BE + CF$
Thus, perimeter of the triangle is greater than the sum of altitudes

Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

ABCD is a quadrilateral. Then which of the following is true? 

  1. $\displaystyle AC+BC<(AB+BC+CD+DA)$
  2. $\displaystyle AC+BD<\frac { 1 }{ 2 } \left( AB+BC+CD+DA \right) $
  3. $\displaystyle AC+BD>\frac { 1 }{ 4 } \left( AB+BC+CD+DA \right) $
  4. $\displaystyle AC+BD<\frac { 1 }{ 4 } \left( AB+BC+CD+DA \right) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\displaystyle AB+BC>AC$ (considering $\displaystyle \Delta ABC$)
$\displaystyle BC+CD>BD$ (considering $\displaystyle \Delta BCD$)
$\displaystyle CD+DA>AC$ (considering $\displaystyle \Delta ADC$)
$\displaystyle DA+AB>BD$ (considering $\displaystyle \Delta ABD$)
Adding all four inequalities, we get
$\displaystyle 2(AB+BC+CD+DA)>2(AC+BD)$
$\displaystyle AB+BC+CD+DA>AC+BD$

Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

Two sides of an acute-angled triangle are $6\ cm$ and $2\ cm$ respectively. Which one of the following represents the correct range of the third side in cm?

  1. $(4, 8)$
  2. $(4, 2\sqrt {10})$
  3. $(4\sqrt {2}, 8)$
  4. $(4\sqrt {2}, 2\sqrt {10})$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If two sides of the triangle are 'a' and 'b' units, then the range of the third side is $(a - b, a + b)$.
So the range of the third side is $(4, 8)$.

Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

In a $\Delta ABC$, which of the following relation is correct where A, B, C are the vertices and D, E, F are the corresponding mid-points?

  1. $AD + BE + CF < AB + BC + CA$
  2. $AB + BE + CF < AD + BC + CA$
  3. $AD + BC + CF < AB + BE + CA$
  4. $AD + BE + CF > AB + BC + CA$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a $\Delta ABC$ let perimeter equal to $AB + BC + CA$ and sum of altitudes is equal to $AD + BE + CF$. Now since $AD\bot BC$
$\therefore AD < AB, AD < AC$
$\therefore AD + AD < AB + AC$
$ 2AD < AB + AC$
$\therefore BE \bot CA$
$\Rightarrow BE < BC, BE < BA$
$\Rightarrow BE + BE < BC + BA$
$2BE < BC +BA.........(2)$
$CF \bot AB$
$\therefore CF < CA, CF < CB$
$\therefore CF + CF < CA + CB$
$2CF < CA + CB.......(3)$
On adding (1), (2) and (3), we get
$2(AD+BE+CF)< AB+ AC + BC+ BA + CA + CB < 2AB + 2BC + 2 CA < 2(AB+BC+CA)$
Hence $AD + BE + CF < AB + BC + CA$