Tag: triangle inequality

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Questions Related to triangle inequality

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

Two sides of a triangle have lengths $7$ and $9$. Which of the following could not be the length of the third side?

  1. $4$
  2. $5$
  3. $7$
  4. $11$
  5. $16$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

An important rule to remember about triangles is called the third side rule: the length of the third side of a triangle is less than the sum of the lengths of the other two sides and greater than the (positive) difference of the lengths of the other two sides. 

For this triangle, the length of the third side must be greater than $9-7=2$97=2 and less than $9+7=16$9+7=16. All the answers are possible except for answer E, which is equal to $16$16 but not less than $16$16.

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

Which of the following sets of measurements can be used to construct a triangle?

  1. $4\ cm, 5\ cm, 6\ cm$
  2. $4\ cm, 3\ cm, 8\ cm$
  3. $5\ cm, 6\ cm, 12\ cm$
  4. $6\ cm, 3\ cm, 10\ cm$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Because the sum of the length of any $2$ sides of the triangle should be greater than the third side, which is only satisfied by option 1.

$(4+5)>6$
$(4+6)>5$
$(5+6)>4$

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 $\Delta ABC, AD \bot BC; BE \bot AC; CF \bot AB.$ then which of the following option is correct:

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

In triangle ADB,
$AD + BD > AB$
In triangle ADC,
$AD + DC > AC$
In tringle BEC,
$BE + EC > BC$
In tringle BEA,
$BE + AE > AB$
In traingle, CFA,
$CF + FA > AC$
In traingle, CFB,
$CF + FB > BC$
Add all the inequalities,
\$2(AD + BE + CF) + AB + BC + CA > 2(AB + BC+ CA)$
\$2 (AD + BE + CF) > AB + BC + CA$
or $AD + BE + CF < AB + BC + CA$

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 $AD + BE + CF < AB + BC + CA$

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

The longest side of a triangle is three times the shortest side and the third side is $2$ cm shorter than the longest side. If the perimeter of the triangle is at least $61$ cm, find the minimum length of the shortest-side.

  1. $9$ cm
  2. $11$ cm
  3. $16$ cm
  4. $61$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the shortest side $= s$
Longest side $= 3s$
Third side $= 3s -2$
Now perimeter $= s + 3s +3s -2 = 7s -2 \ge 61$
$\therefore 7s \ge 63$
$\therefore s \ge 9$
Thus minimum length of the shortest side$= 9$ cm

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

If the inequality $\left( m-2 \right) { x }^{ 2 }+8x+m+4>0$ is satisfied for all $x\epsilon R$, then least integral m is

  1. $4$
  2. $5$
  3. $6$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$(m-2){ x }^{ 2 }+8x+m+4>0$
${ b }^{ 2 }-4ac>0$
${ (8) }^{ 2 }-4(m-2)(m+4)>0$
$64-4({ m }^{ 2 }-2m-8)>0$
$64-4{ m }^{ 2 }+8m+32>0$
$-4{ m }^{ 2 }+8m+96>0$
$-{ m }^{ 2 }+2m+24>0$
$-{ m }^{ 2 }-6m+4m+24>0$
$-m(m+6)+4(m+6)>0$
$(4-m)(m+6)>0$
The least integral of m is $4$.
Multiple choice maths congruence and inequalities of triangles inequalities of a triangle triangle inequality inequalities in triangle

Triangle ABC has integral sides AB, BC measuring $2001$ unit and $1002$ units respectively. Then the number of such triangles, is?

  1. $3002$
  2. $2003$
  3. $1003$

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

Let the sides be a=2001, b=1002, and c be the third side. By the triangle inequality, |a-b| < c < a+b. So 999 < c < 3003. The number of possible integer values for c is 3003 - 999 - 1 = 2003.