integrated math 2 - semester 1 final exam review 1
(use this packet to study - every topic and difficulty level mirrors the actual final)
section 1: similarity (12 problems - calculators ok)
1. a dilation maps a side of length 4 to a side of length 18. type of dilation: __________ scale factor: __________
2. preimage sides: 9, 12, 15 image sides: 6, 8, x find x and state whether it is a reduction or enlargement. justify type of dilation.
3. two similar triangles have scale factor 5/3. a smaller side is 12. corresponding larger side = ______
4. scale factor = 3/8. the larger rectangle side = 40 cm. the smaller corresponding side = ______
5. what theorems do we use to prove similar triangles? which of these theorems do we mostly use in application problems?
6. what are the 4 types of congruent angle pairs that we use to help justify that theorem from problem #4?
7. mirror problem: a student walks 18 m from a flagpole, places mirror, walks back 1.4 m past the mirror, eye height 1.65 m. height of flagpole ≈ ______ m (round to hundredth)
8. justify (using congruence theorems) the two pairs of congruent angles needed to use aa similarity to prove triangles in #7 similar.
9. shadow problem: maya is 5.5 ft tall, casts 6 ft shadow. at the same time, a tree’s shadow ends at the same point and is 33 ft long total from maya’s feet. tree height = ______ ft
10. two similar triangles, scale factor 4/5. area of larger = 80 cm². area of smaller = ______ cm²

Question

integrated math 2 - semester 1 final exam review 1
(use this packet to study - every topic and difficulty level mirrors the actual final)
section 1: similarity (12 problems - calculators ok)
1. a dilation maps a side of length 4 to a side of length 18. type of dilation: __________ scale factor: __________
2. preimage sides: 9, 12, 15  image sides: 6, 8, x find x and state whether it is a reduction or enlargement. justify type of dilation.
3. two similar triangles have scale factor 5/3. a smaller side is 12. corresponding larger side = ______
4. scale factor = 3/8. the larger rectangle side = 40 cm. the smaller corresponding side = ______
5. what theorems do we use to prove similar triangles? which of these theorems do we mostly use in application problems?
6. what are the 4 types of congruent angle pairs that we use to help justify that theorem from problem #4?
7. mirror problem: a student walks 18 m from a flagpole, places mirror, walks back 1.4 m past the mirror, eye height 1.65 m. height of flagpole ≈ ______ m (round to hundredth)
8. justify (using congruence theorems) the two pairs of congruent angles needed to use aa similarity to prove triangles in #7 similar.
9. shadow problem: maya is 5.5 ft tall, casts 6 ft shadow. at the same time, a tree’s shadow ends at the same point and is 33 ft long total from maya’s feet. tree height = ______ ft
10. two similar triangles, scale factor 4/5. area of larger = 80 cm². area of smaller = ______ cm²

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Determine dilation type
Since the image side (18) is longer than the pre - image side (4), the dilation is an enlargement.
## Step2: Calculate scale factor
The scale factor $k$ of a dilation is given by the ratio of the length of the image side to the length of the pre - image side. So $k=\frac{18}{4}=\frac{9}{2} = 4.5$
# Answer:
Type of dilation: Enlargement; Scale factor: $\frac{9}{2}$ (or $4.5$)

### Problem 2:
# Explanation:
## Step1: Find the scale factor
Take the ratio of corresponding sides. For the sides 9 and 6, the scale factor $k=\frac{6}{9}=\frac{2}{3}$. For the sides 12 and 8, $k = \frac{8}{12}=\frac{2}{3}$.
## Step2: Find $x$
Using the scale factor $\frac{2}{3}$, we have $\frac{x}{15}=\frac{2}{3}$. Cross - multiply: $3x=15\times2$, so $x = 10$.
## Step3: Determine dilation type
Since the scale factor $\frac{2}{3}<1$, it is a reduction. We justify it because the length of each image side is less than the length of the corresponding pre - image side.
# Answer:
$x = 10$; Type of dilation: Reduction; Justification: Scale factor $\frac{2}{3}<1$

### Problem 3:
# Explanation:
## Step1: Recall the scale factor relationship
If the scale factor of two similar triangles is $k=\frac{\text{length of larger side}}{\text{length of smaller side}}$, here $k = \frac{5}{3}$ and the smaller side $s = 12$.
## Step2: Calculate the larger side
Let the larger side be $L$. Then $\frac{L}{12}=\frac{5}{3}$. Cross - multiply: $3L=12\times5$, so $L = 20$
# Answer:
$20$

### Problem 4:
# Explanation:
## Step1: Recall the scale factor formula
The scale factor $k=\frac{\text{length of smaller side}}{\text{length of larger side}}$, here $k=\frac{3}{8}$ and the larger side $L = 40$ cm. Let the smaller side be $S$.
## Step2: Solve for $S$
We have $\frac{S}{40}=\frac{3}{8}$. Cross - multiply: $8S=40\times3$, so $S = 15$ cm.
# Answer:
$15$ cm

### Problem 5:
# Brief Explanations:
The theorems used to prove similar triangles are AA (Angle - Angle), SAS (Side - Angle - Side), and SSS (Side - Side - Side) similarity theorems. In application problems, we mostly use the AA (Angle - Angle) similarity theorem because it is often easier to identify two pairs of congruent angles (for example, in problems involving parallel lines, right angles, or angles of elevation/depression) than to identify proportional sides.
# Answer:
Theorems: AA, SAS, SSS; Mostly used: AA

### Problem 6:
# Brief Explanations:
The four types of congruent angle pairs used to justify the AA (Angle - Angle) similarity theorem are: 1. Vertical angles (they are congruent). 2. Corresponding angles (formed by a transversal cutting parallel lines, they are congruent). 3. Alternate interior angles (formed by a transversal cutting parallel lines, they are congruent). 4. Right angles (all right angles are congruent).
# Answer:
Vertical angles, Corresponding angles, Alternate interior angles, Right angles

### Problem 7:
# Explanation:
## Step1: Set up the proportion
We can use similar triangles. Let $h$ be the height of the flagpole. The ratio of the height of the student to the distance from the student to the mirror (beyond the mirror) is equal to the ratio of the height of the flagpole to the distance from the flagpole to the mirror. So $\frac{1.65}{1.4}=\frac{h}{18 + 1.4}$
## Step2: Solve for $h$
First, calculate the denominator on the right: $18 + 1.4=19.4$. Then cross - multiply: $1.4h=1.65\times19.4$. $1.65\times19.4 = 31.91$. Then $h=\frac{31.91}{1.4}\approx22.79$
# Answer:
$22.79$ m

### Problem 8:
# Brief Explanations:
In the mirror problem (problem 7), we have two right triangles. One triangle is formed by the student's height, the distance from the student to the mirror (beyond the mirror). The other triangle is formed by the flagpole's height and the distance from the flagpole to the mirror. The right angles of the two triangles are congruent (all right angles are congruent). Also, the angles of incidence and reflection (formed at the mirror) are congruent (by the law of reflection, which gives us a pair of congruent angles). So we have two pairs of congruent angles (AA), which justifies the similarity of the two triangles.
# Answer:
The two right angles of the triangles are congruent (right angles are congruent), and the angles of incidence and reflection at the mirror are congruent (by the law of reflection), so we have AA similarity.

### Problem 9:
# Explanation:
## Step1: Set up the proportion
Let $h$ be the height of the tree. The triangle formed by Maya's height and her shadow is similar to the triangle formed by the tree's height and its shadow. Maya's shadow is 6 ft, the tree's shadow is $33 - 6=27$ ft? Wait, no. Wait, the tree's shadow ends at the same point as Maya's shadow. Maya's height is 5.5 ft, her shadow is 6 ft. The tree's shadow length is 33 ft (from Maya's feet to the end of the tree's shadow). So the ratio of height to shadow length is the same. So $\frac{5.5}{6}=\frac{h}{33}$
## Step2: Solve for $h$
Cross - multiply: $6h=5.5\times33$. $5.5\times33 = 181.5$. Then $h=\frac{181.5}{6}=30.25$
# Answer:
$30.25$ ft

### Problem 10:
# Explanation:
## Step1: Recall the area ratio of similar figures
If the scale factor of two similar figures is $k$, the ratio of their areas is $k^{2}$. Here the scale factor $k=\frac{4}{5}$, so the ratio of the area of the smaller triangle to the area of the larger triangle is $(\frac{4}{5})^{2}=\frac{16}{25}$
## Step2: Let the area of the smaller triangle be $A$
We know that $\frac{A}{80}=\frac{16}{25}$. Cross - multiply: $25A=80\times16$. $80\times16 = 1280$. Then $A=\frac{1280}{25}=51.2$
# Answer:
$51.2$ $cm^{2}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Determine dilation type

Step2: Calculate scale factor

Step1: Find the scale factor

Step2: Find \(x\)

Step3: Determine dilation type

Step1: Recall the scale factor relationship

Step2: Calculate the larger side

Answer:

Problem 2: