Find the number of positive three-digit even integers whose digits are among 9, 8, 7, 5, 3, and 1.
2 answers:
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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 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.
therefore, .
