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HiSET: Math : Special triangles

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Example Question #1 : 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?

Possible Answers:

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

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

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

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

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

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 .

2x=2

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

As you can see, for this particular triangle, x=1 . 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 x=1 , 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 x=1 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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Example Question #1 : Special Triangles

30 60 90

Examine the above triangle. Which of the following correctly gives the area of $\bigtriangleup ABC$ ?

Possible Answers:

None of the other choices gives the correct response.

$32 \sqrt{3}$

$32 \sqrt{2}$

Correct answer:

$32 \sqrt{3}$

Explanation:

Since $\angle A$ is a right angle - that is, $m \angle A = 90$ - and $m \angle C = 30$ , it follows that

$m \angle B = 180 - (m \angle A + m \angle C) = 180 - (90+30) = 60$ ,

making $\bigtriangleup ABC$ a 30-60-90 triangle.

By the 30-60-90 Triangle Theorem,

$AB = \frac{1}{2}(BC) = \frac{1}{2}(16) = 8$ ,

and

$AC = AB \sqrt{3} = 8\sqrt{3}$

Refer to the diagram below:

30 60 90

The area of a right triangle is equal to half the product of the lengths of its legs, so

$a = \frac{1}{2}(AB)(AC) = \frac{1}{2}(8)(8 \sqrt{3})= 32 \sqrt{3}$ ,

the correct response.

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Example Question #1 : Special Triangles

30 60 90

Examine the above triangle. Which of the following correctly gives the perimeter of $\bigtriangleup ABC$ ?

Possible Answers:

$15+5\sqrt{2}$

$10+10 \sqrt{3}$

$15+5\sqrt{3}$

Correct answer:

$15+5\sqrt{3}$

Explanation:

Since $\angle A$ is a right angle - that is, $m \angle A = 90$ - and $m \angle C = 30$ , it follows that

$m \angle B = 180 - (m \angle A + m \angle C) = 180 - (90+30) = 60$ ,

making $\bigtriangleup ABC$ a 30-60-90 triangle.

By the 30-60-90 Triangle Theorem,

$AB = \frac{1}{2}(BC) = \frac{1}{2}(10) = 5$ ,

and

$AC = AB \sqrt{3} = 5\sqrt{3}$

Refer to the diagram below:

30 60 90

The perimeter - the sum of the sidelengths - is

$P = 10 + 5 + 5 \sqrt{3} = 15 + 5 \sqrt{3}$ .

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HiSET: Math : Special triangles

Study concepts, example questions & explanations for HiSET: Math

All HiSET: Math Resources

Example Questions

Example Question #1 : Special Triangles

Example Question #1 : Special Triangles

Example Question #1 : Special Triangles

All HiSET: Math Resources

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