The front of a house is in the shape of a triangle. Suppose the base is 1 meter les than its height and the area of the triangle is 15 m^2. What is the height of the triangle.

Answer 1

The area of a triangle is (base/2) x height, or (b/2) x h. For this triangle, the base is one meter less than the height, so b = h-1. Substitute h-1 for b in the area formula to get an equation with just one variable:

A = (b/2) x h = (h-1)/2 x h

Distributing h across the binomial (h-1) gives

A = (h² - h)/2, now multiply both sides by 2:

2A = h² - h, substitute 15 for A

2(15) = h² - h, or 30 = h² -h

Subtract 30 from both sides to get a quadratic equation in standard form:

0 = h² - h - 30

Now use the quadratic formula to solve for h:

h = (-b +/- √(b2 - 4ac)) / 2a, where a = 1, b = -1, and c = -30

b2 = 1, and 4ac = -120, so (b2 - 4ac) = 1 - (-120) = 121

Substituting:

h = (1 +/- √(121)) / 2

h = (1 +/- 11)/2

h = 12/2 = 6, or h = -10/2 = -5

But since the height cannot be negative, h = 6, and the base is 1 less than that,

So h = 6 and b = 5

Hope that helps

Answer 2

Hi Aiden!

The area of the triangle is given by the formula A = 1/2 * base * height.

They've given us that the area is 15, so we know that A= 15, or rewriting as below,

A = 1/2 * base * height

15 = 1/2 * base * height

30 = base * height.

But base = height - 1, so we find that 30 = (height - 1) * height. You might see by inspection that this has the solution height = 6, giving 30 = (6-1)*6 = 5*6, or you might want to rewrite and use algebra. See below.

30 = (height - 1) * height

30 = (h-1)*h

30 = h² - h

0 = h² - h - 30, and this factors as 0 = (h - 6)(h + 5), giving two solutions h = 6(as above) and a nonsensical solution h = -5, which we discard recalling that h stood for the height of the triangle.

So the height is 6 meters.

Hope that helps!

I'll be happy to elaborate to make it clearer, so I'd appreciate any feedback. Thank you!