triangle question

Question

Suppose a triangle has an area of 10ft^2. If the base of the triangle is 1 foot longer than the height of the triangle, what are the lengths of the base and height?

 

 

How do i solve this?

Answer 1

Recall that the area formula of a triangle is:

 

A = bh/2 ("base times height divided by two")

 

Since you know the area, you may substitute it in place of A at any time, however that still leaves two unknowns, b and h.  The good news is that you know their relationship to each other, which you can write as an equation.

 

"the base of the triangle is 1 foot longer than the height" translates to:

b = h + 1

 

Making that substitution into the area formula yields:

 

A = (h+1)h/2

 

Distributing the h gives:

 

A = (h^2 + h)/2

 

Substitute the known area:

 

10 = (h^2 + h)/2

 

Multiply both sides by two to clear out the fraction on the right and use the symmetric property to flip the equation:

 

h^2 + h = 20

 

Subtract 20 from both sides and you have the quadratic:

 

h^2 + h - 20 = 0

 

Which is factorable.  Since the leading coefficient on h^2 is 1, we just need two numbers that multiply to the constant, -20, but add up to the middle coefficient (the one on h), which is 1.  These are -4 and 5.

 

Therefore h^2 + h - 20 = (h + 5)(h - 4) = 0

 

This gives us the following solutions for h:

h + 5 = 0 -> h = -5

h - 4 = 0 -> h = 4

 

Since we cannot have a negative number for the height of a triangle, we discard h = -5.

 

Now returning to our equation that expresses the base in terms of height:

 

b = h + 1

b = 4 + 1

b = 5

 

The triangle has a base of 5ft and a height of 4ft.

Answer 2

There are a couple ways to answer this.

 

Easy way:

start with the formula for area: 10 = 1/2 B H

Multiply both sides by 2 to get 20 = B H

So the question is , what 2 numbers multiply to 20 where 1 is a more than the other?

1/20, 2/10, 4/5 <-

 

Hard way:

Use both equations and substitute:

10 = 1/2 B H

B = H + 1

 

10 = 1/2 (H+1) H

20 = H^2 + H

0 = H^2 + H - 20

which factors to -5 and 4 for the H and since a line can't be negative, we know H = 4 and B = 5