Mastery Check Solutions

Show What You Know

Note

These formulas are a derivation of the Pythagorean Theorem and the ways to verify types of triangles.

Triangles can be classified by their angle measure as acute, right, or obtuse. If the side lengths of a triangle are known, the angle classification can be determined by these formulas:

acute
$c < \sqrt{a^{2} + b^{2}}$

right
$c = \sqrt{a^{2} + b^{2}}$

obtuse
$c > \sqrt{a^{2} + b^{2}}$

Show all work to determine the value of x that will form a right triangle for the given figure. Write the solution in simplified radical form.

$\begin{array}{rcl} c & = & \sqrt{a^{2} + b^{2}} \\ 20 & = & \sqrt{x^{2} + x^{2}} \\ {(20)}^{2} & = & {(\sqrt{2 x^{2}})}^{2} \\ 400 & = & 2 x^{2} \\ 200 & = & x^{2} \\ \sqrt{x^{2}} & = & \sqrt{200} = \sqrt{2 \cdot 100} \\ x & = & 10 \sqrt{2} \end{array}$

Note

Note that you only need the principal root because side lengths cannot be negative.

Q: Why is it only necessary to write the principal root as the solution?

A: Because you cannot have a negative side length.

Using the figure, determine the values of x that will form an acute triangle.

$\begin{array}{rcl} c & < & \sqrt{a^{2} + b^{2}} \\ x + 8 & < & \sqrt{x^{2} + {(x + 7)}^{2}} \\ {(x + 8)}^{2} & < & {(\sqrt{x^{2} + {(x + 7)}^{2}})}^{2} \\ x^{2} + 16 x + 64 & < & x^{2} + {(x + 7)}^{2} \\ 16 x + 64 & < & x^{2} + 14 x + 49 \\ 0 & < & x^{2} - 2 x - 15 \\ 0 & < & (x + 3) (x - 5) \\ \begin{matrix} x + 3 > 0 & x - 5 > 0 \\ x > - 3 & x > 5 \end{matrix} \end{array}$

To form an acute triangle $x > 5$

Note

The value of x must be larger than zero (rather than – 3) because side lengths of figures are greater than zero.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Solve for the values that make the principal root true. (Name the restrictions on the radicand.)
Solve a radical inequality.
Graph the solution to a radical inequality on a number line.