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Design Problem: If the height of a triangle is five inches less than the length of its base, and if the area of the triangle is 52 square inches, find the base and the height.

User H Bellamy
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Final answer:

The problem is solved by setting up an equation using the area of a triangle formula, which then is manipulated to derive a quadratic equation. The positive solution for the base is then used to determine the height, which is five inches less than the base.

Step-by-step explanation:

Finding the Base and Height of a Triangle Given Its Area

To solve the design problem where the height of a triangle is five inches less than the length of its base, and the area of the triangle is 52 square inches, we can use the formula for the area of a triangle which is Area = 1/2 × base × height.

Let's define the base of the triangle as b inches. According to the problem, the height (h) will then be b - 5. Plugging these values into the formula for the area, we get:

52 = 1/2 × b × (b - 5)

To solve for b, we first multiply both sides by 2 to get rid of the fraction:

104 = b × (b - 5)

Next, distribute b across the expression (b - 5):

104 = b^2 - 5b

This leaves us with a quadratic equation. Setting it equal to zero:

b^2 - 5b - 104 = 0

We then factor or use the quadratic formula to find the value of b. Assuming we get a positive value for the base, we plug it back into h = b - 5 to find the height.

Following these steps will provide the dimensions of the base and height of the triangle.

User Akela
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