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Zorluk: OrtaTriangles: Properties, Perimeter, and Area

A triangle has a base of length 1010 and an area of 3030. Which of the following statements could be true about this triangle? Select all that apply.

  1. The altitude corresponding to the base of length 1010 is 66.Cevap
  2. The triangle is a right triangle.Cevap
  3. C
    The perimeter of the triangle is 2222.
  4. The perimeter of the triangle is 3030.Cevap
  5. E
    The perimeter of the triangle is 2424.

Cevap

The altitude perpendicular to the given base must be 6, the triangle can be a right triangle, and the perimeter can be equal to 30.
The area formula directly forces the altitude to be 6. Setting the altitude at one endpoint of the base constructs a valid right triangle. Furthermore, the minimum perimeter of any triangle with base 10 and height 6 is 10+261≈25.6210 + 2\sqrt{61} \approx 25.62, so any perimeter value greater than or equal to 25.6225.62 (such as 30) is attainable.

Adım Adım Çözüm

1
Calculate the required altitude of the triangle.
Using Area=12×base×h\text{Area} = \frac{1}{2} \times \text{base} \times h, 30=12(10)h  ⟹  h=630 = \frac{1}{2}(10)h \implies h = 6.
The area and base are fixed, determining a unique height.
2
Evaluate whether the triangle can be a right triangle.
A right triangle with legs 1010 and 66 has area 12×10×6=30\frac{1}{2} \times 10 \times 6 = 30.
Choosing the altitude to meet the base at an endpoint creates a right angle.
3
Determine the lower bound for the perimeter of the triangle.
The third vertex lies on a line parallel to the base at a distance of 66. By symmetry, the minimum sum of the remaining two sides occurs when the triangle is isosceles with base 1010 split into two segments of length 55. Each equal side is 52+62=61≈7.8102\sqrt{5^2 + 6^2} = \sqrt{61} \approx 7.8102. The minimum perimeter is 10+261≈25.6210 + 2\sqrt{61} \approx 25.62.
The shortest path from two fixed base endpoints to a parallel line is formed when the reflection creates equal angles, making the triangle isosceles.
4
Assess the possible perimeter values based on the lower bound.
Perimeters 2222 and 2424 are strictly below the minimum boundary of ≈25.62\approx 25.62 and are impossible. Since the perimeter can take any value in [10+261,∞)[10 + 2\sqrt{61}, \infty), a perimeter of 3030 is possible.
Continuous movement of the top vertex increases the side lengths smoothly without bound.

Anahtar Kavram

Triangle area formula Area=12bh\text{Area} = \frac{1}{2}bh and geometric optimization of perimeter for a given base and height.
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Konular

Triangles: Properties, Perimeter, and Area

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