Question

Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure below shows some of the dimensions but is not drawn to scale. Is the shaded triangle a right triangle?

Answer 1

To determine whether the shaded triangle is a right triangle, we need to use the fact that the three right triangles together form a rectangle.

Let's label some of the dimensions as shown in the diagram:

[asy]

unitsize(0.25cm);

pair A = (0,0), B = (10,0), C = (8,6), D = (6,0), E = (8,0), F = (10,6), G = (8,2);

draw(A--B--F--C--cycle);

draw(D--E);

draw(C--G);

label("$10$", (A+B)/2, S);

label("$6$", (C+F)/2, N);

label("$8$", (B+E)/2, E);

label("$2$", (C+G)/2, E);

label("?", (D+G)/2, NE);

label("$?$", (G+E)/2, S);

fill(C--G--D--cycle, gray(0.7));

[/asy]

The dimensions of the rectangle are 12 units by 14 units, so we know that:

\begin{align*}

10 + 8 &= 18 \\

6 + 2 + ? &= 14

\end{align*}

Simplifying the first equation, we get $8 = 14 - 6$. Substituting this into the second equation, we get:

$$2 + ? = 8$$

Therefore, the length of the shaded side of the triangle is 6 units. Now we can use the Pythagorean theorem to determine whether the shaded triangle is a right triangle. Letting $x$ be the length of the other leg of the shaded triangle, we have:

\begin{align*}

x^2 + 6^2 &= 8^2 \\

x^2 + 36 &= 64 \\

x^2 &= 28 \\

x &= 2\sqrt{7}

\end{align*}

Since $2\sqrt{7}$ is not an integer, we can conclude that the shaded triangle is not a right triangle.