practice & problem solving
leveled practice in 7 and 8, find the missing side length of each triangle.
7.
the length of the hypotenuse is (square) units.
8.
the length of leg (b) is about (square) inches.
9. what is the length of the hypotenuse of the triangle when (x = 2)?
10. what is the length of side (a)?
11. use the pythagorean theorem to find the unknown side length of the right triangle.
12. what is the length of the unknown leg of the right triangle rounded to the nearest tenth of a foot?

Question

Accepted Answer

### Problem 7
# Explanation:
## Step1: Identify legs and hypotenuse
In a right triangle, the Pythagorean theorem is $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs, and $c$ is the hypotenuse. Here, the legs are 9 and 12, so we substitute $a = 9$ and $b = 12$ into the formula: $9^2 + 12^2 = c^2$
## Step2: Calculate the squares
Calculate $9^2 = 81$ and $12^2 = 144$, so the equation becomes $81 + 144 = c^2$
## Step3: Sum the values
Add 81 and 144: $81 + 144 = 225$, so $225 = c^2$
## Step4: Take the square root
Take the square root of both sides: $\sqrt{225} = \sqrt{c^2}$, which gives $c = 15$
# Answer:
The length of the hypotenuse is $\boldsymbol{15}$ units.

### Problem 8
# Explanation:
## Step1: Identify legs and hypotenuse
Using the Pythagorean theorem $a^2 + b^2 = c^2$, here the hypotenuse $c = 8$ in and one leg $a = 6$ in, so the formula is $6^2 + b^2 = 8^2$
## Step2: Calculate the squares
Calculate $6^2 = 36$ and $8^2 = 64$, so the equation becomes $36 + b^2 = 64$
## Step3: Solve for $b^2$
Subtract 36 from both sides: $b^2 = 64 - 36 = 28$
## Step4: Take the square root
Take the square root of 28: $b=\sqrt{28}\approx5.3$ (rounded to one decimal place)
# Answer:
The length of leg $b$ is about $\boldsymbol{5.3}$ inches.

### Problem 9
# Explanation:
## Step1: Substitute $x = 2$ into the leg expressions
First, find the lengths of the legs. For the leg $4x + 4$, substitute $x = 2$: $4(2)+4 = 8 + 4 = 12$. For the leg $3x - 1$, substitute $x = 2$: $3(2)-1 = 6 - 1 = 5$
## Step2: Apply the Pythagorean theorem
Using $a^2 + b^2 = c^2$ with $a = 12$ and $b = 5$, we get $12^2 + 5^2 = c^2$
## Step3: Calculate the squares
Calculate $12^2 = 144$ and $5^2 = 25$, so $144 + 25 = c^2$
## Step4: Sum and take the square root
Sum: $144 + 25 = 169$, then $\sqrt{169}=c$, so $c = 13$
# Answer:
The length of the hypotenuse is $\boldsymbol{13}$ units.

### Problem 10
# Explanation:
## Step1: Identify legs and hypotenuse
In the right triangle, the hypotenuse $c = 39$ cm and one leg is $36$ cm. Let the unknown leg be $a$. Using the Pythagorean theorem $a^2 + 36^2 = 39^2$
## Step2: Calculate the squares
Calculate $36^2 = 1296$ and $39^2 = 1521$, so the equation becomes $a^2 + 1296 = 1521$
## Step3: Solve for $a^2$
Subtract 1296 from both sides: $a^2 = 1521 - 1296 = 225$
## Step4: Take the square root
Take the square root: $a=\sqrt{225}=15$
# Answer:
The length of side $a$ is $\boldsymbol{15}$ cm.

### Problem 11
# Explanation:
## Step1: Identify legs and hypotenuse
The hypotenuse $c = 26$ m and one leg $a = 24$ m. Let the unknown leg be $b$. Using the Pythagorean theorem $24^2 + b^2 = 26^2$
## Step2: Calculate the squares
Calculate $24^2 = 576$ and $26^2 = 676$, so the equation becomes $576 + b^2 = 676$
## Step3: Solve for $b^2$
Subtract 576 from both sides: $b^2 = 676 - 576 = 100$
## Step4: Take the square root
Take the square root: $b=\sqrt{100}=10$
# Answer:
The length of the unknown side is $\boldsymbol{10}$ m.

### Problem 12
# Explanation:
## Step1: Identify legs and hypotenuse
The hypotenuse $c = 9$ ft and one leg $a = 2$ ft. Let the unknown leg be $b$. Using the Pythagorean theorem $2^2 + b^2 = 9^2$
## Step2: Calculate the squares
Calculate $2^2 = 4$ and $9^2 = 81$, so the equation becomes $4 + b^2 = 81$
## Step3: Solve for $b^2$
Subtract 4 from both sides: $b^2 = 81 - 4 = 77$
## Step4: Take the square root and round
Take the square root: $b=\sqrt{77}\approx8.8$ (rounded to the nearest tenth)
# Answer:
The length of the unknown leg is $\boldsymbol{8.8}$ feet.

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QUESTION IMAGE

Question

Explanation:

Problem 7

Step1: Identify legs and hypotenuse

Step2: Calculate the squares

Step3: Sum the values

Step4: Take the square root

Step1: Identify legs and hypotenuse

Step2: Calculate the squares

Step3: Solve for \(b^2\)

Step4: Take the square root

Step1: Substitute \(x = 2\) into the leg expressions

Step2: Apply the Pythagorean theorem

Step3: Calculate the squares

Step4: Sum and take the square root

Answer:

Problem 8