Tag: congruence and inequalities of triangles

Quizzes (1) Questions (47)

Questions Related to congruence and inequalities of triangles

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 perimeter of a triangle is $.........$ than the sum of its medians.

  1. Greater

  2. Lesser

  3. Equal

  4. May be greater or lesser
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

We know that sum of any two sides of the triangle is greater than twice the median bisecting the third side 
Hence, $AB + AC > 2 AD$ (1) 
$AB + BC > 2 BE$ (2)
$BC + AC > 2 CF$ (3)
Adding the three equations:
\$2 (AB + BC + AC) > 2 (AD + BE + CF)$
$AB + BC + AC > AD + BE + CF$
Hence, the perimeter of the triangle is greater than the sum of the medians.

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