SAT Right Triangles Practice

The correct answer is 12. The diagonal of a rectangle forms a right triangle, where the shorter side and the longer side of the rectangle are the legs of the triangle and the diagonal of the rectangle is the hypotenuse of the triangle.

It's given that the length of the rectangle's diagonal is $3\sqrt{17}$ and the length of the rectangle's shorter side is 3. Thus, the length of the hypotenuse of the right triangle formed by the diagonal is $3\sqrt{17}$ and the length of one of the legs is 3.

By the Pythagorean theorem, if a right triangle has a hypotenuse with length $c$ and legs with lengths $a$ and $b$, then $a^2+b^2=c^2$. Substituting $3\sqrt{17}$ for $c$ and 3 for $b$ in this equation yields $a^2+(3{)}^2={\left(3\sqrt{17}\right)}^2$, or $a^2+9=153$.

Subtracting 9 from both sides of this equation yields $a^2=144$. Taking the square root of both sides of this equation yields $a=\pm \sqrt{144}$, or $a=\pm 12$.

Since $a$ represents a length, which must be positive, the value of $a$ is 12. Thus, the length of the rectangle's longer side is 12.

The correct answer is $\frac{16}{23}$.

In a right triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

In the triangle shown, the length of the side opposite the angle with measure $x^{\circ }$ is 16 units and the length of the hypotenuse is 23 units.

Therefore, the value of $\sin x^{\circ }$ is $\frac{16}{23}$.

Note that 16/23, .6956, .6957, 0.695, and 0.696 are examples of ways to enter a correct answer.

A rectangle's diagonal divides a rectangle into two congruent right triangles, where the diagonal is the hypotenuse of both triangles. It's given that the length of the diagonal is $5\sqrt{17}$ and the length of the rectangle's shorter side is 5.

Therefore, each of the two right triangles formed by the rectangle's diagonal has a hypotenuse with length $5\sqrt{17}$, and a shorter leg with length 5. To calculate the length of the longer leg of each right triangle, the Pythagorean theorem, $a^2+b^2=c^2$, can be used, where $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypotenuse of the triangle.

Substituting 5 for $a$ and $5\sqrt{17}$ for $c$ in the equation $a^2+b^2=c^2$ yields $5^2+b^2={\left(5\sqrt{17}\right)}^2$, which is equivalent to $25+b^2=25(17)$, or $25+b^2=425$. Subtracting 25 from each side of this equation yields $b^2=400$.

Taking the positive square root of each side of this equation yields $b=20$. Therefore, the length of the longer leg of each right triangle formed by the diagonal of the rectangle is 20.

It follows that the length of the rectangle's longer side is 20.