Answer:
- side: 17 inches
- altitude: 6 inches
Step-by-step explanation:
The formula for the area of a triangle gives a relationship between area, height, and base length. The problem statement gives the area, and it gives another relation between height and base length. Using these two relations, you can solve for the dimensions of the triangle.
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setup
The side to which the altitude is measured is called the "base" in the area formula. The altitude is called the "height." The relation between them given by the area formula is ...
A = 1/2bh
51 = 1/2bh . . . . . . using the given value for area
The other relation given by the problem statement is that the base is 1 less than 3 times the height:
b = 3h -1 . . . . . . . base is 1 less than 3 times altitude
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solution
Using this second equation to substitute for 'b' in the first equation, we have ...
51 = 1/2(3h -1)h
102 = 3h² -h . . . . multiply by 2, eliminate parentheses
3h² -h -102 = 0 . . . . . put in standard form
3h² -18h +17h -102 = 0 . . . . . prepare to factor
3h(h -6) +17(h -6) = 0 . . . . . factor pairs of terms
(3h +17)(h -6) = 0 . . . . . . . . finish factoring
Solutions are the values of h that make these factors zero:
3h +17 = 0 ⇒ h = -17/3
h -6 = 0 ⇒ h = 6
We know that dimensions in a geometry problem must be positive, so the solution here is h = 6. Then b = 3h -1 = 3(6) -1 = 17. The dimensions are all in inches.
The length of the side is 17 inches; the altitude to that side is 6 inches.
