Question

1. find the perimeter.

9x + 8, 12x - 8, 12x - 8 (triangle sides)

1. find the perimeter and the area.

x + 3, x, 15 (right triangle sides)

Explanation:

Problem 1: Find the perimeter of the isosceles triangle

Step1: Recall perimeter formula

Perimeter of a triangle is the sum of all its sides. The triangle has sides \( 9x + 8 \), \( 12x - 8 \), and \( 12x - 8 \).

Step2: Sum the side lengths

Add the three sides:
\( (9x + 8) + (12x - 8) + (12x - 8) \)

Step3: Combine like terms

• For \( x \)-terms: \( 9x + 12x + 12x = 33x \)
• For constant terms: \( 8 - 8 - 8 = -8 \)

So, the perimeter is \( 33x - 8 \).

Problem 4: Find the perimeter and area of the right triangle

First, use the Pythagorean theorem to find \( x \), then calculate perimeter and area.

Step1: Apply Pythagorean theorem

In a right triangle, \( (leg_1)^2 + (leg_2)^2 = (hypotenuse)^2 \).
Here, \( (x + 3)^2 + x^2 = 15^2 \)

Step2: Expand and simplify

\( x^2 + 6x + 9 + x^2 = 225 \)
\( 2x^2 + 6x + 9 - 225 = 0 \)
\( 2x^2 + 6x - 216 = 0 \)
Divide by 2: \( x^2 + 3x - 108 = 0 \)

Step3: Solve the quadratic equation

Factor: \( (x + 12)(x - 9) = 0 \)
Solutions: \( x = -12 \) (discard, length can’t be negative) or \( x = 9 \)

Step4: Find side lengths

• One leg: \( x = 9 \)
• Other leg: \( x + 3 = 12 \)
• Hypotenuse: \( 15 \) (given)

Step5: Calculate perimeter

Perimeter = \( 9 + 12 + 15 = 36 \)

Step6: Calculate area

Area of a right triangle: \( \frac{1}{2} \times leg_1 \times leg_2 \)
\( \frac{1}{2} \times 9 \times 12 = 54 \)

Answer:

s:

1. Perimeter: \( \boldsymbol{33x - 8} \)
2. Perimeter: \( \boldsymbol{36} \), Area: \( \boldsymbol{54} \)