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Multiple choice maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle
Let $ A(1,2,3), B(0,0,1), C(-1,1,1)$ are the vertices of a $\triangle ABC$. Then, the equation of internal angle bisector through A to side BC is 
  1. $\underset{r}{\rightarrow}=\widehat{i}+2\widehat{j}+3\widehat{k}+\mu (3\widehat{i}+2\widehat{j}+3\widehat{k})$
  2. $\underset{r}{\rightarrow}=\widehat{i}+2\widehat{j}+3\widehat{k}+\mu (3\widehat{i}+4\widehat{j}+3\widehat{k})$
  3. $\underset{r}{\rightarrow}=\widehat{i}+2\widehat{j}+3\widehat{k}+\mu (3\widehat{i}+3\widehat{j}+2\widehat{k})$
  4. $\underset{r}{\rightarrow}=\widehat{i}+2\widehat{j}+3\widehat{k}+\mu (3\widehat{i}+3\widehat{j}+4\widehat{k})$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The internal angle bisector of a triangle divides the opposite side in the ratio of the adjacent sides. The lengths of sides AB and AC can be found to determine the ratio, and then the coordinates of the dividing point on BC are used to find the direction vector of the angle bisector.

Multiple choice maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

In a  $\triangle A B C,$  side  $A B$  has the equation  $2 x + 3 y = 29$  and the side  $A C$  has the equation  $x + 2 y = 16.$  If the mid point of  $B C$  is  $( 5,6 ) ,$  then the equation of  $B C$  is

  1. $2 x + y = 16$
  2. $x + y = 11$
  3. $2 x - y = 4$
  4. $x + y = - 11$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\cfrac { x _{ 1 }+x _{ 2 } }{ 2 } =5\Rightarrow x _{ 1 }+x _{ 2 }=10.....(1)\quad and\quad y _{ 1 }+y _{ 2 }=12.....(2)$

$Point(x _1,y _1)$ lie on line AC
then
$x _1+2y _1=16...(3)$
Similarly $2x _2+3y _2=29....(4)$
$\Rightarrow 2(x _1+x _2)+4y _1+3y _2=32+29\2\times 10+4y _1+36-3y _1=61\y _1=5\Rightarrow x _1=6\Rightarrow x _2=4\ \Rightarrow y _2=7$
now,
we take these two points and make equation,
$AC= x+y=11$

Multiple choice maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

In triangle, three angles are  $x , x + 10 ^ { \circ } + x + 20 ^ { \circ }$  then the biggest is

  1. $70 ^ { \circ }$
  2. $80 ^ { \circ }$
  3. $90 ^ { \circ }$

  4. none
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum of angles in a triangle is 180 degrees. x + (x + 10) + (x + 20) = 180. 3x + 30 = 180, so 3x = 150, x = 50. The angles are 50, 60, and 70 degrees. The biggest is 70 degrees.

Multiple choice maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

In. triangle ABC,$\angle A$ + $\angle B$ = 144 and$\angle A$ + $\angle C$ = 124.
Calculate smallest angle of the triangle.

  1. $36^o$
  2. $56^o$
  3. $46^o$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\angle A + \angle B = 144$...(I)
$\angle A + \angle C = 124$...(II)
In triangle ABC,
$\angle A  + \angle B + \angle C = 180 $
Add, I and II,
$\angle A + \angle B + \angle A + \angle C = 144+ 124$
$180 + \angle A = 268 $
$\angle A = 268 - 180 $
$\angle A = 88$
Put this value in (I)
$\angle A + \angle B = 144$
$88 + \angle B = 144$
$\angle B = 56$
Put this value in (II)
$\angle A + \angle C = 124$
$88 + \angle C = 124$
$\angle C = 36$

Multiple choice maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

If every side of a triangle is doubled, then the area of the new triangle is 'K' times the area of the old one. The value of K is

  1. 2

  2. 3
  3. $\sqrt 2$

  4. 4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Let the area of the triangle be $x$.

We know that the area of the triangle
$=\dfrac{1}{2}\times Height \times Base$
$x=\dfrac{1}{2}\times Height \times Base$              $........ (1)$

According to the question,
$Kx=\dfrac{1}{2}\times 2 \times Height \times 2 \times Base$
$Kx=4\times x$
$K=4$

Hence, this is the answer.
Multiple choice maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

The ratio of the areas of two similar triangles is equal to the

  1. ratio ofcorresponding medians

  2. ratio ofcorresponding sides

  3. ratio of the squares ofcorresponding sides

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

The ratio of the areas of two similar triangles is equal to the square of ratio of their corresponding sides.

Multiple choice decimal representation of rational numbers rational and irrational numbers maths

Number of ordered pairs $(x, y)$ of real numbers satisfying the equation $x^{2} + y^{2} - 24x - 26y + 313 = 0$ is equal to

  1. Infinite

  2. Finite but more than one

  3. Exactly one

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

Rewriting the given equation by completing the square for x and y terms gives (x - 12)^2 + (y - 13)^2 = 0. Since the sum of squares of real numbers can only be zero if each term is individually zero, there is only one unique real solution pair (12, 13). Therefore, the number of ordered pairs is exactly one.