Let us solve the given problems step by step.

6. Two sides of a right triangle are 8” and 12”.

(A) Find the area of the triangle if 8 and 12 are legs.

• Formula for the area of a triangle:
  $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
• Substituting the values $\text{base} = 8$ and $\text{height} = 12$:
  $\text{Area} = \frac{1}{2} \times 8 \times 12 = 48 \, \text{square inches}.$

Answer: $48 \, \text{square inches}$.

(B) Find the area of the triangle if 8 and 12 are a leg and the hypotenuse.

• Using the Pythagorean Theorem to find the missing leg:
  $a^2 + b^2 = c^2, \quad \text{where } c = 12 \text{ and } a = 8.$ Solving for $b$:
  $b^2 = c^2 - a^2 = 12^2 - 8^2 = 144 - 64 = 80.$ $b = \sqrt{80} = 4\sqrt{5} \, \text{inches}.$
• Now calculate the area:
  $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 4\sqrt{5} = 16\sqrt{5} \, \text{square inches}.$

Answer: $16\sqrt{5} \approx 35.8 \, \text{square inches}$.

7. The area of a square is $81 \, \text{cm}^2$. Find the perimeter of the square.

• The formula for the area of a square:
  $\text{Area} = \text{side}^2.$ Solving for the side length:
  $\text{side} = \sqrt{81} = 9 \, \text{cm}.$
• The formula for the perimeter of a square:
  $\text{Perimeter} = 4 \times \text{side}.$ Substituting $\text{side} = 9$:
  $\text{Perimeter} = 4 \times 9 = 36 \, \text{cm}.$

Answer: $36 \, \text{cm}$.

8. An isosceles triangle has congruent sides of 20 cm. The base is 10 cm. What is the area of the triangle?

• The height of the triangle can be found using the Pythagorean Theorem. Let the height be $h$, and note that the base is split into two equal parts of $5 \, \text{cm}$:
  $h^2 + 5^2 = 20^2.$ Solving for $h$:
  $h^2 = 20^2 - 5^2 = 400 - 25 = 375, \quad h = \sqrt{375} = 5\sqrt{15} \, \text{cm}.$
• The area of the triangle:
  $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 5\sqrt{15} = 25\sqrt{15} \, \text{square cm}.$

Answer: $25\sqrt{15} \approx 96.6 \, \text{square cm}$.

Would you like a detailed explanation of any step?

5 Related Questions:

1. How can we generalize the Pythagorean Theorem for non-right triangles?
2. What is the difference between exact answers and approximations in geometry?
3. How can you derive the formula for the area of an isosceles triangle?
4. What is the relationship between the diagonal and the side length of a square?
5. Can the triangle inequality theorem determine if sides form a triangle?

Tip: Always double-check units in calculations to ensure consistency!