At the crossroads of algebra and geometry

Geometry Level 1

The lengths of the sides of a triangle are $x$ cm, $(x+1)$ cm, and $(x+2)$ cm. Determine $x$ so that this triangle is a right-angled triangle.

The answer is 3.

2 solutions

Soha Farhin Pine Pine
Feb 1, 2017

Relevant wiki: Pythagorean Theorem

According to Pythagorean Theorem,

$x^2+(x+1)^2=(x+2)^2$

$x^2=(x+2)^2-(x+1)^2$

$x^2=(x+2+x+1)(x+2-x-1)$

$x^2=2x+3$

You end up with a quadratic equation:

$x^2-2x-3=0$

Split the middle term:

$x^2-3x+x-3=0$

Solve for x by factoring:

$(x-3)(x+1)=0$

$\therefore x=3$ $\text{or}$ $x=-1$

As length cannot be negative, the second solution is not applicable.

So, the answer must be $\boxed{x=3}$ .

Might I add that the solutions to the equation are $x = -1, 3$ , but since the sides of a triangle must be positive, $x = 3$ is the only possible $x$ .

Zach Abueg - 4 years, 4 months ago

Thanks. Now that I think of it - my solution looks incomplete without these vital points.

Soha Farhin Pine Pine - 4 years, 4 months ago

A Former Brilliant Member
Feb 1, 2017

At the crossroads of algebra and geometry

The answer is 3.

2 solutions

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