Answer :
Sure, let's work through the problem of determining the height and area of Mr. Anderson's isosceles triangular roof using the given base length of 14 feet. Here we will assume that the lengths of the other two congruent sides are known to be 10 feet each.
### Step-by-step solution:
1. Identify and Understand the Triangle:
- You have an isosceles triangle with a base length (`b`) of 14 feet.
- The lengths of the two congruent sides (`s`) are both 10 feet.
2. Split the Triangle:
- Divide the isosceles triangle into two right triangles by drawing a perpendicular line from the apex to the midpoint of the base. This line represents the height (`h`) of the isosceles triangle.
- The base is 14 feet, so each half of the base is 7 feet.
3. Apply the Pythagorean Theorem:
- In the right triangle, the hypotenuse (`s`) is 10 feet, the base (`b/2`) is 7 feet, and the height (`h`) can be found using the Pythagorean theorem: [tex]\( s^2 = \left(\frac{b}{2}\right)^2 + h^2 \)[/tex].
- Substitute the known values: [tex]\( 10^2 = 7^2 + h^2 \)[/tex].
- Simplify: [tex]\( 100 = 49 + h^2 \)[/tex].
- Solve for `h`: [tex]\( h^2 = 100 - 49 \)[/tex], [tex]\( h^2 = 51 \)[/tex], so [tex]\( h = \sqrt{51} \)[/tex].
4. Calculate the Height:
- [tex]\( h \approx 7.141428 \)[/tex] feet.
5. Calculate the Area:
- The area `A` of a triangle is given by: [tex]\( A = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex].
- Substitute the known values: [tex]\( A = \frac{1}{2} \times 14 \times 7.141428 \)[/tex].
- Simplify: [tex]\( A \approx 49.989999 \)[/tex] square feet.
### Conclusion:
- The height of Mr. Anderson’s triangular-shaped roof is approximately [tex]\( 7.141428 \)[/tex] feet.
- The area of the roof is approximately [tex]\( 49.989999 \)[/tex] square feet.
These calculations help Mr. Anderson understand the dimensions and area of the triangular roof he plans to build for his shed.
### Step-by-step solution:
1. Identify and Understand the Triangle:
- You have an isosceles triangle with a base length (`b`) of 14 feet.
- The lengths of the two congruent sides (`s`) are both 10 feet.
2. Split the Triangle:
- Divide the isosceles triangle into two right triangles by drawing a perpendicular line from the apex to the midpoint of the base. This line represents the height (`h`) of the isosceles triangle.
- The base is 14 feet, so each half of the base is 7 feet.
3. Apply the Pythagorean Theorem:
- In the right triangle, the hypotenuse (`s`) is 10 feet, the base (`b/2`) is 7 feet, and the height (`h`) can be found using the Pythagorean theorem: [tex]\( s^2 = \left(\frac{b}{2}\right)^2 + h^2 \)[/tex].
- Substitute the known values: [tex]\( 10^2 = 7^2 + h^2 \)[/tex].
- Simplify: [tex]\( 100 = 49 + h^2 \)[/tex].
- Solve for `h`: [tex]\( h^2 = 100 - 49 \)[/tex], [tex]\( h^2 = 51 \)[/tex], so [tex]\( h = \sqrt{51} \)[/tex].
4. Calculate the Height:
- [tex]\( h \approx 7.141428 \)[/tex] feet.
5. Calculate the Area:
- The area `A` of a triangle is given by: [tex]\( A = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex].
- Substitute the known values: [tex]\( A = \frac{1}{2} \times 14 \times 7.141428 \)[/tex].
- Simplify: [tex]\( A \approx 49.989999 \)[/tex] square feet.
### Conclusion:
- The height of Mr. Anderson’s triangular-shaped roof is approximately [tex]\( 7.141428 \)[/tex] feet.
- The area of the roof is approximately [tex]\( 49.989999 \)[/tex] square feet.
These calculations help Mr. Anderson understand the dimensions and area of the triangular roof he plans to build for his shed.
