Question
A triangle is to have a base \( \frac{1}{3} \) as long as its height. Its area must be 6 square feet. How long will its base be a. 1 foot b. 1.5 feet c. 2 feet d. 2.5 feet e. 3 feet
Ask by Young Park. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The base of the triangle is 2 feet.
Solution
Let the height of the triangle be h and the base be b. According to the problem, b = (1/3)h. The area A of a triangle is given by:
A = (1/2) × base × height
We are given A = 6 square feet, so:
(1/2) × b × h = 6
Substitute b with (1/3)h:
(1/2) × (1/3)h × h = 6
(1/6)h² = 6
Multiply both sides by 6:
h² = 36
Taking the square root:
h = 6 (only the positive value is considered because h represents a length).
Now, find b:
b = (1/3)h = (1/3) × 6 = 2
Thus, the base of the triangle is 2 feet. The correct answer is option (c).
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Simplify this solution Beyond the Answer
To find the base of the triangle, we use the area formula for a triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Let the height be \( h \). Since the base is \( \frac{1}{3} \) of the height, we can express the base as \( \frac{h}{3} \). Now, substituting into the area formula: \[ 6 = \frac{1}{2} \times \frac{h}{3} \times h \] This simplifies to: \[ 6 = \frac{h^2}{6} \] Multiplying both sides by 6 gives: \[ 36 = h^2 \] Taking the square root results in: \[ h = 6 \text{ feet} \] Now, we can find the base: \[ \text{base} = \frac{h}{3} = \frac{6}{3} = 2 \text{ feet} \] So, the answer is c. 2 feet.
