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Questions Related to maths

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

The lengths of two sides of a triangle are $3 $ cm and $4 $ cm. Which of the following, can be the length of third side to form a triangle?

  1. $0.5 $ cm
  2. $5 $ cm
  3. $8 $ cm
  4. $10$ cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

(I) We know that $(3 + 4) $ cm is greater than third side.
Thus, the third side is smaller than $7$ cm
(ii) we know that $(4 - 3) $ cm is smaller than third side .
Thus, the third side is greater than $1 $cm.
Therefore, $1 $ cm < third side $< 7 $ cm.

Thus, $5  $ cm can be the length of third side for a triangle. 

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

Find all possible lengths of the third side, if sides of a triangle have $3$ and $9$.

  1. $6 < x < 12$
  2. $5 < x < 12$
  3. $6 < x < 10$
  4. $6 < x < 11$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Triangle Inequality theorem states that the sum of any $2$ sides of a triangle must be greater than the measure of the third side.
So, difference of two sides $< x <$ sum of two sides, will give you the possible length of a triangle.
Therefore, $9 - 3 < x < 9 + 3$
$6 < x < 12$ is the possible length of the third side of a triangle.
For checking the possible length: Take $3, 9, 7$
$3 + 9 > 7 (a + b > c)$
$9 + 7 > 3 (b + c > a)$
$3 + 7 > 9 (a + c > b)$
Which satisfy the triangle inequality theorem.

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

The construction of a triangle $ABC$, given that $BC =$ $6$ cm, $B =$ $45 ^{\circ}$ is not possible when difference of $AB$ and $AC$ is equal to:

  1. $6.9$ cm
  2. $5.2$ cm
  3. $5.0$ cm
  4. $4.0$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

According to the theorem of inequalities, the sum of any two sides of the triangle is greater than the third side.

Therefore, $AC+BC>AB$
$\Rightarrow BC>AB-AC$
Therefore, only the first option that is $6.9$ cm does not satisfy the above equation. Rest all the options satisfy the equation.

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

In triangle ABC, (b+c) cos A+(c+a)cos B+(a+b)cos C is equal to

  1. $0$
  2. $1$
  3. $a+b+c$
  4. $2(a+b+c)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(b+c) \cos A+(c+a)\cos B+(a+b)\cos C$


$\Rightarrow$  $b\cos A+c\cos A+c\cos B+a\cos B+a\cos C+b\cos C$

$\Rightarrow$  $(b\cos C+c\cos B)+(c\cos A+a\cos C)+(a\cos B+b\cos A)$  ----( 1 )
Using projection formula,
$a=(b\cos C+c\cos B)$
$b=(c\cos A+a\cos C)$
$c=(a\cos B+b\cos A)$
Substituting above values in ( 1 ) we get,
$\Rightarrow$  $a+b+c$
$\therefore$   $(b+c) \cos A+(c+a)\cos B+(a+b)\cos C=a+b+c$

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

Find all possible lengths of the third side, if sides of a triangle have $2$ and $5$.

  1. $2 < x < 7$
  2. $3 > x < 7$
  3. $3 < x > 7$
  4. $3 < x < 7$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Triangle Inequality theorem states that the sum of any $2$ sides of a triangle must be greater than the measure of the third side.
So, difference of two sides $< x <$ sum of two sides, will give you the possible length of a triangle.
Therefore, $5 - 2 < x < 5 + 2$
$3 < x < 7$ is the possible length of the third side of a triangle.
For checking the possible length: Take $2, 5, 4$
$2 + 5 > 4 (a + b > c)$
$5 + 4 > 2 (b + c > a)$
$2 + 4 > 5 (a + c > b)$
Hence, the above condition satisfied the triangle inequality theorem.

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

A triangle has side lengths of $6$ inches and $9$ inches. If the third side is an integer, calculate the minimum possible perimeter of the triangle (in inches).

  1. $4$
  2. $15$
  3. $8$
  4. $19$
  5. $29$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the third side be $x$.
Sum of any two sides of a triangle is greater than the third side. 

Hence, $6+x>9$ or $x>3$ and $6+9>x$ or $x<15$.
Therefore, $x\epsilon (3,15)$
Hence, the minimum possible integral value of $x$ is $4$. 
Thus the minimum possible length of the third side is $4$. 
Hence, the minimum possible perimeter is $4+6+9=19$ units.

Multiple choice maths triangle inequality construction of parallel lines and triangles triangle inequality related to lines and triangles sum of the lengths of two sides of a triangle

Which statement is true about the difference of any two sides of a triangle?

  1. It is greater than the third side

  2. It is zero

  3. It is lesser than the third side

  4. It is lesser than zero
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $a,b,c$ be the sides of triangle.

For constructing a triangle sum of any two sides must be greater than third side
$\Rightarrow a+b>c$
$\Rightarrow a>c-b$
$\Rightarrow c-b<a.......(i)$
Also $a+c>b$
$\Rightarrow  c>a-b$
$\Rightarrow a-b>c.....(ii)$
Also $b+c>a$
$\Rightarrow c>a-b$
$\Rightarrow a-b>c.......(iii)$
From $(i),(ii)$ and $(iii)$ it is clear that difference of any two sides is greater than the third side.
So option $C$ is correct.

Multiple choice maths numbers and place value forming numbers formation of greatest and smallest numbers identifying the largest and smallest numbers with given digits

Smallest 6-digit number that can be formed by the digits 9, 6, 0, 5, 8, 1 is

  1. $015,689$
  2. $1,05,689$
  3. $5,01,689$
  4. $9,86,510$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Rightarrow$  The given numbers are $9,\,6,\,0,\,5,\,8,\,1$

$\Rightarrow$  To form smallest 6-digit number, we have to start number with smallest digit and end with largest digit.
$\Rightarrow$  Here, we can not use $0$ as first digit because then number will becomes 5 digit.
$\therefore$   The smallest 6-digit number = $1,05,689$.

Multiple choice maths numbers and place value forming numbers formation of greatest and smallest numbers identifying the largest and smallest numbers with given digits

Smallest 6-digit number that can be formed by the digits 9, 6, 0, 5, 8, 1 is 

  1. 0,15,689

  2. 1,05,689

  3. 5,01,689

  4. 9,86,510
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Smallest  6  digit number that can be formed by the digits 9,6,0,5,8,1 is  1,05,689.  

 0 cannot  in the first place . Then it will become 5 digit number.