Certainly! Let's address each part of the question with a detailed explanation.
### Part a: Draw a Right Triangle with 3 in and 5 in Perpendicular Sides
Imagine a right triangle where the two legs (perpendicular sides) are 3 inches and 5 inches long. For this triangle:
- One leg (side [tex]\(a\)[/tex]) is 3 inches.
- The other leg (side [tex]\(b\)[/tex]) is 5 inches.
- The right angle is between these two legs.
### Part b: Find the Perimeter of the Triangle
To find the perimeter of the triangle, we need to know the lengths of all three sides. We already know two sides: 3 inches and 5 inches. The third side, the hypotenuse ([tex]\(c\)[/tex]), can be found using the Pythagorean theorem:
#### Step 1: Apply the Pythagorean Theorem
The Pythagorean theorem states that:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
Substituting the known values:
[tex]\[
c^2 = (3)^2 + (5)^2
\][/tex]
[tex]\[
c^2 = 9 + 25
\][/tex]
[tex]\[
c^2 = 34
\][/tex]
#### Step 2: Calculate the Hypotenuse
To find [tex]\(c\)[/tex], take the square root of both sides:
[tex]\[
c = \sqrt{34}
\][/tex]
This simplifies to approximately:
[tex]\[
c \approx 5.830951894845301
\][/tex]
So, the hypotenuse is approximately 5.831 inches.
#### Step 3: Calculate the Perimeter
The perimeter of a triangle is the sum of the lengths of all its sides:
[tex]\[
\text{Perimeter} = a + b + c
\][/tex]
Substituting the values:
[tex]\[
\text{Perimeter} = 3 + 5 + 5.830951894845301
\][/tex]
[tex]\[
\text{Perimeter} \approx 13.8309518948453
\][/tex]
So, the perimeter of the right triangle is approximately 13.831 inches.
### Summary
1. The right triangle has legs of 3 inches and 5 inches.
2. The hypotenuse is approximately 5.831 inches.
3. The perimeter of the triangle is approximately 13.831 inches.
