Question

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Answer 1

Answer:

Easy... Given the triangle RST with Coordinates  R(2,1), S(2, -2) and T(-1 , -2).

A dilation is a transformation which produces an image that is the same shape as original one, but is different size.  

Since, the scale factor  is greater than 1, the image is enlargement or a stretch.  

Now, draw the dilation image of the triangle RST with center (2,-2) and scale factor

Since, the center of dilation at S(2,-2) is not at the origin, so the point S and its image  are same.

Now, the distances from the center of the dilation at point S to the other points R and T.  

The dilation image will be of each of these distances,

, so =5 ;

, so   

Now, draw the image of RST i.e R'S'T'

Since,  [By using hypotenuse of right angle triangle] and .

2)

(a)

Disagree with the given statement.

Side Angle Side postulate (SAS) states that:

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle then these two triangles are congruent.

Given: B is the midpoint of  i.e

In the triangle ABD and triangle CBD, we have

  (SIDE)            [Given]

  (SIDE)            [Reflexive post]

Since, there is no included angle in these triangles.

∴  is not congruent to  .

Therefore, these triangles does not follow the SAS congruence postulates.

(b)

SSS(SIDE-SIDE-SIDE) states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Since it is also given that  .

therefore, in the triangle ABD and triangle CBD, we have

  (SIDE)            [Given]

  (SIDE)           [Given]

  (SIDE)            [Reflexive post]

therefore by, SSS postulates .

3)

Given that:   are vertical angles, as they are formed by intersecting lines.

Therefore

, by the definition of linear pairs

and  and   and  are linear pair.

By linear pair theorem, and    are supplementary,  and   are supplementary.

Equate the above expressions:

Subtract the angle 2 from both sides in the above expressions

∴

By Congruent Supplement theorem: If two angles are supplements of the same angle, then the two angles are congruent.

Answer 2

Answer:

1)

Given the triangle RST with Coordinates  R(2,1), S(2, -2) and T(-1 , -2).

A dilation is a transformation which produces an image that is the same shape as original one, but is different size.  

Since, the scale factor $\frac{5}{3}$ is greater than 1, the image is enlargement or a stretch.  

Now, draw the dilation image of the triangle RST with center (2,-2) and scale factor $\frac{5}{3}$

Since, the center of dilation at S(2,-2) is not at the origin, so the point S and its image $S{}'$ are same.

Now, the distances from the center of the dilation at point S to the other points R and T.  

The dilation image will be$\frac{5}{3}$ of each of these distances,

$SR=3$, so $S{}'R{}'$=5 ;

$ST=3$, so $S{}'T{}'=5$  

Now, draw the image of RST i.e R'S'T'

Since, $RT=3\sqrt{2}$ [By using hypotenuse of right angle triangle] and $R{}'T{}'=5\sqrt{2}$.

2)

(a)

Disagree with the given statement.

Side Angle Side postulate (SAS) states that:

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle then these two triangles are congruent.

Given: B is the midpoint of $\overline{AC}$ i.e $\overline{AB}\cong \overline{BC}$

In the triangle ABD and triangle CBD, we have

$\overline{AB}\cong \overline{BC}$   (SIDE)            [Given]

$\overline{BD}\cong \overline{BD}$   (SIDE)            [Reflexive post]

Since, there is no included angle in these triangles.

∴ $\Delta ABD$ is not congruent to $\Delta CBD$ .

Therefore, these triangles does not follow the SAS congruence postulates.

(b)

SSS(SIDE-SIDE-SIDE) states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Since it is also given that  $\overline{AD}\cong \overline{CD}$.

therefore, in the triangle ABD and triangle CBD, we have

$\overline{AB}\cong \overline{BC}$   (SIDE)            [Given]

$\overline{AD}\cong \overline{CD}$   (SIDE)           [Given]

$\overline{BD}\cong \overline{BD}$   (SIDE)            [Reflexive post]

therefore by, SSS postulates $\Delta ABD\cong \Delta CBD$.

3)

Given that:  $\angle1=\angle 3$ are vertical angles, as they are formed by intersecting lines.

Therefore

, by the definition of linear pairs

$\angle 1$ and $\angle 2$ and $\angle 3$  and $\angle 2$ are linear pair.

By linear pair theorem, $\angle 1$ and $\angle 2$   are supplementary, $\angle 2$ and $\angle 3$  are supplementary.

$m\angle1+m\angle 2=180^{\circ}$

$m\angle2+m\angle 3=180^{\circ}$

Equate the above expressions:

$m\angle 1+m\angle 2=m\angle 2+m\angle 3$

Subtract the angle 2 from both sides in the above expressions

∴$m\angle 1=m\angle 3$

By Congruent Supplement theorem: If two angles are supplements of the same angle, then the two angles are congruent.

therefore, $\angle 1\cong \angle 3$.