2-51: plot triangle abc with vertices a(0,0), b(3,4), and c(3,0) on graph paper. using the origin as the point of dilation, enlarge it by a factor of 2 (imagine using two rubber bands). label this new triangle abc
a) what are the side lengths of the original triangle, δabc?
b) what are the side lengths of the enlarged triangle, δabc?
c) calculate the area and the perimeter of δabc.
2-52: rewrite each statement below into conditional (\if ..., then ...\) form. then state whether it is true or false. if it is false, modify the conclusion to make the statement true
a) the area of a rectangle with base x and height 2x is 2x².
b) the perimeter of a rectangle with base x and height 3y is 3xy.
c) a rectangle with base 2 feet and height 3 feet has an area of 864 square inches.
2-53: the area of the triangle at right is 25 square feet. determine the value of h. then calculate the perimeter of the entire triangle. show all work.

Question

Accepted Answer

### 2 - 51
#### a)
# Explanation:
## Step1: Calculate length of AB
Use distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For $A(0,0)$ and $B(3,4)$, $AB=\sqrt{(3 - 0)^2+(4 - 0)^2}=\sqrt{9 + 16}=\sqrt{25}=5$.
## Step2: Calculate length of BC
For $B(3,4)$ and $C(3,0)$, $BC=\sqrt{(3 - 3)^2+(0 - 4)^2}=\sqrt{0+16}=4$.
## Step3: Calculate length of AC
For $A(0,0)$ and $C(3,0)$, $AC=\sqrt{(3 - 0)^2+(0 - 0)^2}=3$.
# Answer:
$AB = 5$, $BC = 4$, $AC = 3$

#### b)
# Explanation:
When a figure is dilated by a factor of $k = 2$, the side - lengths of the new figure are $k$ times the side - lengths of the original figure.
## Step1: Find $A'B'$
$A'B'=2\times AB=2\times5 = 10$.
## Step2: Find $B'C'$
$B'C'=2\times BC=2\times4 = 8$.
## Step3: Find $A'C'$
$A'C'=2\times AC=2\times3 = 6$.
# Answer:
$A'B' = 10$, $B'C' = 8$, $A'C' = 6$

#### c)
# Explanation:
## Step1: Calculate area of $\triangle A'B'C'$
Since $\triangle A'B'C'$ is a right - triangle (because $A'C'$ is horizontal and $B'C'$ is vertical), area formula $A=\frac{1}{2}\times base\times height$. Here, base $A'C' = 6$ and height $B'C' = 8$, so $A=\frac{1}{2}\times6\times8=24$.
## Step2: Calculate perimeter of $\triangle A'B'C'$
Perimeter $P=A'B'+B'C'+A'C'=10 + 8+6=24$.
# Answer:
Area = 24 square units, Perimeter = 24 units

### 2 - 52
#### a)
# Explanation:
The conditional form is: If a rectangle has a base of length $x$ and a height of length $2x$, then its area is $2x^{2}$.
The formula for the area of a rectangle is $A = base\times height$. Here, $A=x\times2x = 2x^{2}$, so it is true.
# Answer:
Conditional form: If a rectangle has a base of length $x$ and a height of length $2x$, then its area is $2x^{2}$. True.

#### b)
# Explanation:
The conditional form is: If a rectangle has a base of length $x$ and a height of length $3y$, then its perimeter is $3xy$.
The formula for the perimeter of a rectangle is $P=2(base + height)=2(x + 3y)=2x+6y$. The given statement is false.
To make it true, the conclusion should be: then its perimeter is $2x + 6y$.
# Answer:
Conditional form: If a rectangle has a base of length $x$ and a height of length $3y$, then its perimeter is $3xy$. False. Modified conclusion: then its perimeter is $2x + 6y$.

#### c)
# Explanation:
First, convert feet to inches. 1 foot = 12 inches, so a base of 2 feet is $2\times12 = 24$ inches and a height of 3 feet is $3\times12=36$ inches.
The conditional form is: If a rectangle has a base of 24 inches and a height of 36 inches, then its area is 864 square inches.
The area of the rectangle $A=24\times36 = 864$ square inches, so it is true.
# Answer:
Conditional form: If a rectangle has a base of 24 inches and a height of 36 inches, then its area is 864 square inches. True.

### 2 - 53
# Explanation:
## Step1: Find the value of $h$
The area formula for a triangle is $A=\frac{1}{2}\times base\times height$. Given $A = 25$ square feet, base $b=4$ feet.
We have $25=\frac{1}{2}\times4\times h$.
Solve for $h$:
$$
\begin{align*}
25&=\frac{1}{2}\times4\times h\
25&=2h\
h& = 12.5
\end{align*}
$$
## Step2: Calculate the perimeter of the triangle
Using the Pythagorean theorem to find the length of the third side $l$ of the right - triangle. $l=\sqrt{4^{2}+12.5^{2}}=\sqrt{16 + 156.25}=\sqrt{172.25}\approx13.12$.
Perimeter $P=4 + 6+13.12=23.12$ feet.
# Answer:
$h = 12.5$ feet, Perimeter $\approx23.12$ feet

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QUESTION IMAGE

Question

Explanation:

2 - 51

a)

Step1: Calculate length of AB

Step2: Calculate length of BC

Step3: Calculate length of AC

Step1: Find $A'B'$

Step2: Find $B'C'$

Step3: Find $A'C'$

Step1: Calculate area of $\triangle A'B'C'$

Step2: Calculate perimeter of $\triangle A'B'C'$

Answer:

b)