Question

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Answer 1

7.5 inches squared

Explanation:

If only distances are provided, there are two methods to find the area of a triangle. Which method to use depends on which lengths are provided:

• Method 1: Base * Height (one leg and the corresponding height)
• Method 2: Heron's Formula (all three legs are known)

Method 1: Base * Height

For any triangle, a line segment from one vertex that terminates perpendicularly to the leg across from it is considered a "height," and the leg that the "height" intersects is "base". If both quantities are known, then the area of the triangle is base times height divided by 2 (sometimes stated as 1/2 times base times height).

In an equation format:
`Area_(triangle)=(1)/(2)base*height`, often abbreviated as
`A_(triangle)=(1)/(2)bh`

For this example, the base at the bottom "6 in" has a corresponding height of "2.5 in", so

`A_(triangle)=(1)/(2)(2.5~\text{in})(6~\text{in})`

`A_(triangle)=7.5\text{ in}^2`

Method 2: Heron's Formula

If you are not provided any of the heights, but only the three legs of the triangle, Heron's formula provides a method of finding the area of the triangle. It is significantly more complicated, and should only be used if one doesn't have one of the heights given.

Heron's formula gives the area of a triangle using the following formula:
`A=√(s(s-a)(s-b)(s-c))` where a, b, and c, are the side lengths of the triangle, and "s" is the "semi-perimeter" (one half of the perimeter) of the triangle:
`s=(a+b+c)/(2)`

To be consistent, let's let a=3 in, b=5 in, and c=6 in.

Calculating the semi-perimeter of the given triangle first:

`s=(a+b+c)/(2)=\frac{(3~\text{in})+(5~\text{in})+(6~\text{in})}{2}=\frac{14~\text{in}}{2}=7~\text{in}`

Calculating the area:

`A=√(s(s-a)(s-b)(s-c))`

`A=\sqrt{(7~\text{in})((7~\text{in})-(3~\text{in}))((7~\text{in})-(5~\text{in}))((7~\text{in})-(6~\text{in}))}`

`A=\sqrt{(7~\text{in})(4~\text{in)(2~\text{in})(1~\text{in})}`

`A=\sqrt{56~\text{in}^4}`

`A=7.48331477355...~\text{in}^2`

`A\approx7.5~\text{in}^2`

Side note: The 2.5 inches that they gave you in the problem isn't exactly 2.5 inches. They rounded to one decimal place there.

Answer 2

Explanation:

A= 1/2 b h b=6 h=2.5

= 7.5