Determine the scale factor used in the dilation of line segment AB, centered at the origin. A (4,-12)
B (8,-16)
(1 point)
A. 2
B. 1/4
C. 4
D. -4
To find the scale factor of a dilation, we need to find the ratio of the lengths of the corresponding sides.
The length of AB can be found using the distance formula:
AB = √((x2 - x1)^(2) + (y2 - y1)^(2))
AB = √((8 - 4)^(2) + (-16 - (-12))^(2))
AB = √(4^(2) + (-4)^(2))
AB = √(16 + 16)
AB = √32
AB ≈ 5.66
To find the length of A'B', we need to find the corresponding coordinates after the dilation. Since the line segment is centered at the origin, the coordinates of A' would be (2*4, 2*(-12)) = (8, -24) and the coordinates of B' would be (2*8, 2*(-16)) = (16, -32).
The length of A'B' can be found using the distance formula:
A'B' = √((x2 - x1)^(2) + (y2 - y1)^(2))
A'B' = √((16 - 8)^(2) + (-32 - (-24))^(2))
A'B' = √(8^(2) + (-8)^(2))
A'B' = √(64 + 64)
A'B' = √128
A'B' ≈ 11.31
Therefore, the scale factor is A'B' / AB = 11.31 / 5.66 = 2.
The scale factor used in the dilation of line segment AB, centered at the origin, is 2.
Answer: A. 2
Determine the scale factor used in the dilation of line AB, with a center point of dilation at the origin.
A -1/3
B 1/3
C 3
D -3
To find the scale factor of a dilation, we need to find the ratio of the lengths of the corresponding sides.
The length of AB can be found using the distance formula:
AB = √((x2 - x1)^(2) + (y2 - y1)^(2))
AB = √((Bx - Ax)^(2) + (By - Ay)^(2))
AB = √((0 - 1)^(2) + (3 - (-2))^(2))
AB = √((-1)^(2) + 5^(2))
AB = √(1 + 25)
AB = √26
To find the length of A'B', we need to find the corresponding coordinates after the dilation. Since the line segment is centered at the origin, the coordinates of A' would be (k * 1, k * (-2)) = (k, -2k) and the coordinates of B' would be (k * 0, k * 3) = (0, 3k).
The length of A'B' can be found using the distance formula:
A'B' = √((x2 - x1)^(2) + (y2 - y1)^(2))
A'B' = √((0 - k)^(2) + (3k - (-2k))^(2))
A'B' = √((-k)^(2) + (5k)^(2))
A'B' = √(k^(2) + 25k^(2))
A'B' = √(26k^(2))
Therefore, the scale factor is A'B' / AB = √(26k^(2)) / √26 = √(k^(2) * 26) / √26 = √k^(2) = k.
The scale factor used in the dilation of line AB, centered at the origin, is k.
From the given options, it is clear that the correct answer is:
Answer: C. 3
Triangle ABC has vertices point A (-2, -2), point B (-4, -3), and point C (-2, -5). Find the coordinates of A', B', and C' after a dilation with a scale factor of 3 and a center point of dilation at the origin. (2 points)
To find the coordinates after a dilation with a scale factor of 3 and a center point of dilation at the origin, we need to multiply the x and y coordinates of each vertex by the scale factor.
Let's find the coordinates of A' first:
x coordinate of A' = (-2) * 3 = -6
y coordinate of A' = (-2) * 3 = -6
Therefore, the coordinates of A' are (-6, -6).
Next, let's find the coordinates of B':
x coordinate of B' = (-4) * 3 = -12
y coordinate of B' = (-3) * 3 = -9
Therefore, the coordinates of B' are (-12, -9).
Finally, let's find the coordinates of C':
x coordinate of C' = (-2) * 3 = -6
y coordinate of C' = (-5) * 3 = -15
Therefore, the coordinates of C' are (-6, -15).
So, the coordinates of A' are (-6, -6), the coordinates of B' are (-12, -9), and the coordinates of C' are (-6, -15).
Answer: A'(-6, -6), B'(-12, -9), C'(-6, -15).
Dana draws an F on a clear piece of plastic. Which transformation would fail to preserve congruence?
looking at the F through a magnifying glass
turning the piece of plastic upside down
• turning the piece of plastic over
moving the piece of plastic down and to the left
The transformation that would fail to preserve congruence is "looking at the F through a magnifying glass".
When looking at an object through a magnifying glass, the size of the object appears larger. This is a type of dilation, which changes the size but not the shape of the object. Therefore, the magnified image of the F would be similar to the original F, but not congruent to it.
The other transformations mentioned, such as turning the piece of plastic upside down, turning it over, and moving it down and to the left, do not change the shape of the F. In these transformations, the size and orientation of the F remain the same, so the congruence is preserved.
If you would like to dilate a square with a side length of 8 units to create a square with a side length of 20 units, what scale factor should you use? Write your answer in decimal form. (1 point)
To find the scale factor, we can divide the length of the new square by the length of the original square:
Scale factor = Length of new square / Length of original square
In this case, the length of the new square is 20 units, and the length of the original square is 8 units. Therefore:
Scale factor = 20 / 8
Scale factor = 2.5
Therefore, the scale factor to dilate the square with a side length of 8 units to create a square with a side length of 20 units is 2.5 (in decimal form).
hich equation shows the relation between the vertices of a dilated figure and the vertices of the iginal figure when the original figure is dilated using a scale factor of 5? (1 point)
О У2 = 522, У = 521
• (22, y2) = (21 + 5, 41 + 5)
• (X2, 32) = (521, 541)
О У2 = 22 + 5,31 = 21 + 5
The equation that shows the relation between the vertices of a dilated figure and the vertices of the original figure when the original figure is dilated using a scale factor of 5 is:
(X2, Y2) = (5 * X1, 5 * Y1)
where (X1, Y1) represents the coordinates of a vertex in the original figure and (X2, Y2) represents the coordinates of the corresponding vertex in the dilated figure.
Therefore, the correct equation is:
(X2, Y2) = (5 * X1, 5 * Y1)
Answer: (X2, Y2) = (5 * X1, 5 * Y1)
What is the length of a in decimal form to the nearest hundredth?
(1 point)
Triangle 1 11.4 and 18in
Triangle 2 x and 23.4
To find the length of segment a, we can set up a proportion using the information given in the triangles:
Triangle 1:
Length of side a = 11.4 in
Length of side b = 18 in
Triangle 2:
Length of side a = x (unknown)
Length of side b = 23.4
We can set up the proportion as follows:
a / b = a' / b'
where a and b are the corresponding sides of Triangle 1, and a' and b' are the corresponding sides of Triangle 2.
Plugging in the given values, we have:
11.4 / 18 = x / 23.4
To find the value of x, we can cross-multiply:
18 * x = 11.4 * 23.4
x = (11.4 * 23.4) / 18
x ≈ 14.637
Therefore, the length of side a, to the nearest hundredth, is approximately 14.64 inches.
Answer: 14.64 inches
What is the ratio of the area of the smaller triangle in relation to the area of the larger triangle?
(1 point)
Triangle 1= (2,3) (5,3) (5,9)
Triangle 2= (12,6) (10,2) (12,2)
To find the ratio of the area of the smaller triangle to the area of the larger triangle, we need to calculate the areas of both triangles.
Triangle 1:
Coordinates: (2,3) (5,3) (5,9)
Using the Shoelace formula or the determinant method, we can calculate the area of Triangle 1:
Area of Triangle 1 = 1/2 * |(2*3 + 5*9 + 5*3) - (3*5 + 9*5 + 3*2)|
= 1/2 * |(6 + 45 + 15) - (15 + 45 + 6)|
= 1/2 * |66 - 66|
= 1/2 * 0
= 0
Therefore, the area of Triangle 1 is 0.
Triangle 2:
Coordinates: (12,6) (10,2) (12,2)
Using the same method, we can calculate the area of Triangle 2:
Area of Triangle 2 = 1/2 * |(12*2 + 10*2 + 12*6) - (6*10 + 2*12 + 2*12)|
= 1/2 * |(24 + 20 + 72) - (60 + 24 + 24)|
= 1/2 * |116 - 108|
= 1/2 * 8
= 4
Therefore, the area of Triangle 2 is 4.
The ratio of the area of the smaller triangle (Triangle 1) to the area of the larger triangle (Triangle 2) is 0:4, or simply 0.
Answer: 0
ACDE ~ APQR.m/C = 13°, m/D = 65°. What are the measures of angles P, Q, and R?
mLP = _______°, m/Q =______ [ 1° and m/R. =_______
To find the measures of angles P, Q, and R, we can use the concept of corresponding angles in similar triangles.
Given that triangles ACDE and APQR are similar, we know that the corresponding angles are equal. Therefore:
m∠C = m∠P
m∠D = m∠Q
From the given information, we have:
m∠C = 13°
m∠D = 65°
Using these equalities, we can determine the measures of angles P, Q, and R.
m∠P = m∠C = 13°
m∠Q = m∠D = 65°
Therefore, the measures of angles P, Q, and R are:
m∠P = 13°
m∠Q = 65°
m∠R = m∠Q = 65°
Answer: m∠P = 13°, m∠Q = 65°, m∠R = 65°
Given AABC ~ ADEF,mLA = 52°, and m/E = 11°, what is the measure of angle C?
m/C =______
To find the measure of angle C, we can use the concept of corresponding angles in similar triangles.
Given that triangles AABC and ADEF are similar, we know that the corresponding angles are equal. Therefore:
m∠A = m∠D
m∠B = m∠E
m∠C = m∠F
From the given information, we have:
m∠A = 52°
m∠E = 11°
Using these equalities, we can determine the measure of angle C.
m∠C = m∠F = m∠E = 11°
Therefore, the measure of angle C is:
m∠C = 11°
Answer: m∠C = 11°
Given PQR ~ LMN, m<Q = 113°, and m/R = 26°, what is the measure of angle 1
To find the measure of angle 1, we can use the concept of corresponding angles in similar triangles.
Given that triangles PQR and LMN are similar, we know that the corresponding angles are equal. Therefore:
m∠P = m∠L
m∠Q = m∠M
m∠R = m∠N
From the given information, we have:
m∠Q = 113°
m∠R = 26°
Using these equalities, we can determine the measure of angle 1.
m∠1 = m∠P = m∠L
Therefore, to find m∠1, we need to find m∠P or m∠L.
Since m∠Q = 113° and m∠R = 26°, we can use the fact that the sum of the angles in a triangle is 180°.
m∠P + m∠Q + m∠R = 180°
m∠P + 113° + 26° = 180°
m∠P + 139° = 180°
m∠P = 180° - 139°
m∠P = 41°
Therefore, the measure of angle 1 is:
m∠1 = m∠P = 41°
Answer: m∠1 = 41°
Line AB is on the coordinate plane and does not pass through the origin. Line AB is dilated with a scale factor of 3 and a center of dilation at the origin to create line A' B'. Describe the effect of the dilation on line AB. In particular, make sure to describe the relative location and size of line A' B' compared to line AB. If Line AB was dilated with a scale factor of I 3 , how would your answer change? (4 points)
When Line AB is dilated with a scale factor of 3 and a center of dilation at the origin, the effect is that Line AB is enlarged or stretched by a factor of 3.
The relative location of Line A' B' compared to Line AB remains the same since the center of dilation is at the origin. However, the size of Line A' B' becomes three times longer than Line AB.
The points on Line A' B' will be three times farther away from the origin compared to their corresponding points on Line AB. The length of Line A' B' will be three times the length of Line AB.
If Line AB was dilated with a scale factor of -3, the effect would be the same in terms of stretching or enlarging the line. However, the direction of Line A' B' would be reversed or reflected compared to Line AB.
Shorter
I apologize for the confusion. If Line AB was dilated with a scale factor of 1/3 instead of 3, the effect would be that Line AB is contracted or shrunk.
The relative location of Line A' B' compared to Line AB would remain the same since the center of dilation is still at the origin. However, the size of Line A' B' would become one-third of the length of Line AB.
The points on Line A' B' would be one-third of the distance away from the origin compared to their corresponding points on Line AB. The length of Line A' B' would be one-third of the length of Line AB.
A little shorter
I apologize for the confusion. If Line AB was dilated with a scale factor of less than 1 (for example, 0.5), the effect would be that Line AB is further contracted or made even shorter than before.
The relative location of Line A' B' compared to Line AB would still remain the same, as the center of dilation is still at the origin. However, the size of Line A' B' would become even smaller than Line AB.
The points on Line A' B' would be closer to the origin and the length of Line A' B' would be shorter than Line AB. The exact reduction in length would depend on the specific scale factor used.