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Flashcards: Special triangles

Two of a triangle's interior angles measure $30^{\circ}$ and $90^{\circ}$ , respectively. If this triangle's hypotenuse is $2\;\textup{in}$ long, what are the lengths of its other sides?

$\frac{1}{2}\;\textup{in}\;\textup{and}\;\sqrt{2}\;\textup{in}$

$\frac{\sqrt2}{2}\;\textup{in}\;\textup{and}\;\sqrt{3}\;\textup{in}$

$\frac{\sqrt3}{2}\;\textup{in}\;\textup{and}\;\sqrt{2}\;\textup{in}$

$\sqrt3\;\textup{in}\;\textup{and}\;1\;\textup{in}$

$\sqrt2\;\textup{in}\;\textup{and}\;2\;\textup{in}$

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Flashcards: Special triangles

Two of a triangle's interior angles measure $30^{\circ}$ and $90^{\circ}$ , respectively. If this triangle's hypotenuse is $2\;\textup{in}$ long, what are the lengths of its other sides?

Your answer:

$\frac{1}{2}\;\textup{in}\;\textup{and}\;\sqrt{2}\;\textup{in}$

Your answer:

$\frac{\sqrt2}{2}\;\textup{in}\;\textup{and}\;\sqrt{3}\;\textup{in}$

Your answer:

$\frac{\sqrt3}{2}\;\textup{in}\;\textup{and}\;\sqrt{2}\;\textup{in}$

Your answer:

$\sqrt2\;\textup{in}\;\textup{and}\;2\;\textup{in}$

Correct answer:

$\sqrt3\;\textup{in}\;\textup{and}\;1\;\textup{in}$

Explanation:

A triangle that has interior angles of $30^{\circ}$ and $90^{\circ}$ is necessarily a 30-60-90 triangle—a special right triangle. We can tell that the third angle about which we're not told anything has to be $60^{\circ}$ because a triangle's interior angles always sum to $180^{\circ}$ , allowing us to solve for the third angle like so:

$a+b+c=180^{\circ}$

$30^{\circ}+b+90^{\circ}=180^{\circ}$

$120^{\circ}+b=180^{\circ}$

$(120^{\circ}+b)-120^{\circ}=(180^{\circ})-120^{\circ}$

$b=60^{\circ}$

Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.

$\frac{30^{\circ}}{x}=\frac{60^{\circ}}{x\sqrt3}=\frac{90^{\circ}}{2x}$

We're told that the hypotenuse of our triangle has a length of $2\;\textup{in}$ . The hypotenuse is the triangle's longest side, so it will be located directly across from its largest angle. In this case, that angle is $90^{\circ}$ . So, we need to set equivalent to $2\;\textup{in}$ and solve for .

$x=\frac{2}{2}=1$

As you can see, for this particular triangle, . Using this information, we can now calculate the lengths of the other sides of the triangle. The side opposite the $30^{\circ}$ angle will be equal to inches; since , this side's length is $1\;\textup{in}$ . The side opposite the $60^{\circ}$ angle will be equal to $x\sqrt3\;\textup{in}$ . Substituting in into this expression, we find that this side has a length of $\sqrt{3}\;\textup{in}$ .

Thus, the correct answer is $\sqrt3\;\textup{in}\;\textup{and}\;1\;\textup{in}$ .

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Flashcards: Special triangles

Flashcards: Special triangles

Flashcards: Special triangles

Flashcards: Special triangles

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Flashcards: Special triangles

Flashcards: Special triangles

Flashcards: Special triangles

Flashcards: Special triangles

Report an issue with this question

DMCA Complaint

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Address

Complaint Details

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