1. in a diagram, △abc has angles 50°, 60°, and 70°. if △def has the same angles, what is true?
a. the triangles are similar.
b. the triangles are not related.
c. the triangles are congruent.
d. the triangles are proportional but not similar.
2. which of the following would prove two triangles are similar using the a.a. (angle-angle) criterion?
a. two sides are congruent.
b. two angles are congruent, and their sides are proportional.
c. two angles are congruent.
d. all three sides are proportional.
3. what does the ratio in a scale, such as 1 cm : 50 m, represent?
a. the relationship between the scale drawing and the actual dimensions.
b. the size of the premise.
c. the angle measures in a scale drawing.
d. the distance between two points on a map.
4. a rectangular garden measures 25 ft × 40 ft. if the scale on a blueprint is 1 in. : 5 ft, what are the dimensions of the garden on the blueprint?
a. 4 in. × 7 in.
b. 6 in. × 9 in.
c. 2.5 in. × 4 in.
d. 5 in. × 8 in.
5. a triangle △jkl with side lengths 4, 6, and 8 is dilated by k = 3. what are the side lengths of the dilated triangle?
a. 4, 6, 8
b. 6, 9, 12
c. 12, 18, 24
d. 12, 18, 24
6. a triangular billboard frame has side lengths 6 meters, 9 meters, and 12 meters. another irregular frame has side lengths 3 meters, 4.5 meters, and 6 meters. are the frames similar?
a. yes, by the sas criterion.
b. no, the sides are not proportional.
c. no, the angles do not match.
d. yes, by the sss criterion.
7. a polygon with one vertex at (5, 2) is dilated by a scale factor of $ \frac{3}{2} $ with the center at (1, 0). what are the coordinates of the dilated vertex?
a. (8, 4)
b. (6, 3)
c. (7.5, 3)
d. (7, 4.5)

Question

1. in a diagram, △abc has angles 50°, 60°, and 70°. if △def has the same angles, what is true?
 a. the triangles are similar.
 b. the triangles are not related.
 c. the triangles are congruent.
 d. the triangles are proportional but not similar.
2. which of the following would prove two triangles are similar using the a.a. (angle-angle) criterion?
 a. two sides are congruent.
 b. two angles are congruent, and their sides are proportional.
 c. two angles are congruent.
 d. all three sides are proportional.
3. what does the ratio in a scale, such as 1 cm : 50 m, represent?
 a. the relationship between the scale drawing and the actual dimensions.
 b. the size of the premise.
 c. the angle measures in a scale drawing.
 d. the distance between two points on a map.
4. a rectangular garden measures 25 ft × 40 ft. if the scale on a blueprint is 1 in. : 5 ft, what are the dimensions of the garden on the blueprint?
 a. 4 in. × 7 in.
 b. 6 in. × 9 in.
 c. 2.5 in. × 4 in.
 d. 5 in. × 8 in.
5. a triangle △jkl with side lengths 4, 6, and 8 is dilated by k = 3. what are the side lengths of the dilated triangle?
 a. 4, 6, 8
 b. 6, 9, 12
 c. 12, 18, 24
 d. 12, 18, 24
6. a triangular billboard frame has side lengths 6 meters, 9 meters, and 12 meters. another irregular frame has side lengths 3 meters, 4.5 meters, and 6 meters. are the frames similar?
 a. yes, by the sas criterion.
 b. no, the sides are not proportional.
 c. no, the angles do not match.
 d. yes, by the sss criterion.
7. a polygon with one vertex at (5, 2) is dilated by a scale factor of $ \frac{3}{2} $ with the center at (1, 0). what are the coordinates of the dilated vertex?
 a. (8, 4)
 b. (6, 3)
 c. (7.5, 3)
 d. (7, 4.5)

Accepted Answer

# 1.
## Brief Explanations:
By the AA (Angle - Angle) similarity criterion, if two triangles have all three angles equal, they are similar. Since $\triangle ABC$ and $\triangle DEF$ have the same angles, they satisfy the AA similarity condition. Congruent triangles require equal side lengths (in addition to equal angles), and if they are similar, they are related. Proportionality is a property of similar triangles.
# Answer:
a. The triangles are similar.

# 2.
## Brief Explanations:
The AA (Angle - Angle) similarity criterion states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Side - side congruence is for congruent triangles (SSS for congruence), side - angle - side proportion is not the AA criterion, and all three sides proportional is the SSS similarity criterion.
# Answer:
c. Two angles are congruent.

# 3.
## Brief Explanations:
A scale ratio, such as $1\mathrm{cm}:50\mathrm{m}$, represents the relationship between the measurements on the scale drawing (or model) and the actual measurements. It is not about the sum of the perimeter (which would be a numerical value calculation), angle measures (which are preserved in similar figures but not what the ratio represents), or a single distance (the ratio is a general relationship).
# Answer:
a. The relationship between the scale drawing and the actual dimensions.

# 4.
## Step - by - Step Format:
### Explanation:
## Step1: Use the scale formula
The scale is $1\mathrm{in}:5\mathrm{ft}$. For the length $l = 25\mathrm{ft}$, if $x$ is the length on the blueprint, then $\frac{x}{25}=\frac{1}{5}$.
## Step2: Solve for $x$ (length)
Cross - multiply: $5x = 25$, so $x=\frac{25}{5}=5\mathrm{in}$.
## Step3: Solve for $y$ (width)
For the width $w = 40\mathrm{ft}$, if $y$ is the width on the blueprint, then $\frac{y}{40}=\frac{1}{5}$. Cross - multiply: $5y = 40$, so $y = 8\mathrm{in}$.
# Answer:
d. $5\mathrm{in}\times8\mathrm{in}$

# 5.
## Step - by - Step Format:
### Explanation:
## Step1: Use the dilation formula
If a triangle with side lengths $a = 4$, $b = 6$, $c = 8$ is dilated by a scale factor $k = 3$, the new side lengths are $a'=k\times a$, $b'=k\times b$, $c'=k\times c$.
## Step2: Calculate new side lengths
$a'=3\times4 = 12$, $b'=3\times6=18$, $c'=3\times8 = 24$.
# Answer:
d. $12,18,24$

# 6.
## Step - by - Step Format:
### Explanation:
## Step1: Check the ratio of side lengths
For the first frame with side lengths $a_1 = 6$, $b_1 = 9$, $c_1 = 12$ and the second frame with side lengths $a_2 = 3$, $b_2 = 4.5$, $c_2 = 6$.
Calculate the ratios: $\frac{a_2}{a_1}=\frac{3}{6}=\frac{1}{2}$, $\frac{b_2}{b_1}=\frac{4.5}{9}=\frac{1}{2}$, $\frac{c_2}{c_1}=\frac{6}{12}=\frac{1}{2}$.
## Step2: Apply SSS similarity criterion
Since $\frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{c_2}{c_1}$, by the SSS (Side - Side - Side) similarity criterion, the two frames are similar.
# Answer:
d. Yes, by the SSS Criterion

# 7.
## Step - by - Step Format:
### Explanation:
## Step1: Use the dilation formula for a point $(x,y)$ with center $(h,k)$
The formula for dilation is $(x',y')=(h + k(x - h),k(y - k)+k)$. Here $h = 0$, $k = 0$ (center $(0,0)$) and the scale factor $s=\frac{3}{2}$, and the point $(x,y)=(5,2)$.
$x'=\frac{3}{2}\times5=\frac{15}{2}=7.5$, $y'=\frac{3}{2}\times2 = 3$.
# Answer:
c. $(7.5,3)$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

1.

Answer:

2.