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Trigonometry : Triangles

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Example Questions

Example Question #1 : 45 45 90 Triangles

In a 45-45-90 triangle, if one leg is . What is the measure of the hypotenuse?

Possible Answers:

$6\sqrt{2}$

$6\sqrt{3}$

$\sqrt{12}$

Correct answer:

$6\sqrt{2}$

Explanation:

One option is to use the Pythagorean Theorem.

Since we have an isosceles triangle, both legs must be congruent.

Plug in to get your answer.

$h=\sqrt{6^2+6^2}$

$h=\sqrt{72}=\sqrt{36\cdot2}=6\sqrt2$

Or, you can remember the 45-45-90 identity, which states that the hypotenuse is $\sqrt{2}$ times the leg.

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Example Question #1 : 45 45 90 Triangles

In a 45-45-90 triangle, if the hypotenuse is 10, what is the perimeter of the triangle?

Possible Answers:

$\sqrt{10}+5\sqrt{2}$

$\frac{10+\sqrt{50}}{3}$

$3\sqrt{10}$

$10\sqrt{2}+10$

Correct answer:

$10\sqrt{2}+10$

Explanation:

Write the Pythagorean Theorem.

a^2+b^2=c^2

In a 45-45-90 triangle, the length of the legs are equal, which indicates that:

a=b

Rewrite the formula and substitute the known sides.

a^2+a^2=c^2

2a^2=10^2

a^2=50

$a=\sqrt{50}=5\sqrt2$

The lengths of the triangle are: $5\sqrt{2},5\sqrt{2},10$

Sum the three lengths for the perimeter.

$10\sqrt{2}+10$

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Example Question #2 : 45 45 90 Triangles

One side of a 45^o - 45^o - 90^o triangle has a length of 3. Which cannot be the length of one of the other sides?

Possible Answers:

$\frac{3}{\sqrt2}$

$3 \sqrt{3}$

$\frac{3 \sqrt2}{2}$

$3\sqrt{2}$

Correct answer:

$3 \sqrt{3}$

Explanation:

If 3 is one of the legs, then the hypotenuse is $3 \sqrt2$ .

If 3 is the hypotenuse, then the legs are $\frac{3}{\sqrt2}$ or equivalently $\frac{3 \sqrt2}{2}$

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Example Question #3 : 45 45 90 Triangles

The hypotenuse of a 45^o - 45^o - 90^o triangle is 4. What is the length of each of the legs?

Possible Answers:

$2 \sqrt{2}$

$4\sqrt{2}$

$\sqrt2$

Correct answer:

$2 \sqrt{2}$

Explanation:

Divide the length of the hypotenuse by $\sqrt{2}$ to get the length of the legs:

$\frac{4 } {\sqrt2} = \frac{4 \sqrt2}{2} = 2 \sqrt2$

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Example Question #1 : 45 45 90 Triangles

The perimeter of a square is 56 feet. What is the length of the diagonal in feet?

Possible Answers:

There is not enough information

$7\sqrt{2}$

$4\sqrt7$

$14\sqrt{2}$

Correct answer:

$14\sqrt{2}$

Explanation:

The perimeter of a square can be found using the formula P=4s , where P is the perimeter and s is the length of the side of the square.

56=4s

s=14

The diagonal of a square forms the hypotenuse of a 45-45-90 triangle, where each leg is the side of the square. In a 45-45-90 triangle, the ratio of the hypotenuse to the leg is $\sqrt2$ , so the diagonal of this square is $14\sqrt2$ .

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Example Question #2 : 45 45 90 Triangles

Find the value of

Possible Answers:

$7\sqrt{2}$

$14\sqrt{2}$

Correct answer:

$14\sqrt{2}$

Explanation:

Solving this problem begins with realizing that all three of our triangles are not only right triangles but isosceles and are therefore 45-45-90 triangles. That means in each triangle to get from the length of a leg to the length of the hypotenuse, we simply multiply by $\sqrt{2}$ . Therefore, the hypotenuse of our bottom triangle is

$7\cdot \sqrt{2}=7\sqrt{2}$

However, the hypotenuse of the bottom triangle is also the leg of the middle triangle. To find the hypotenuse of this triangle, we simply repeat the process.

$7\sqrt2 \cdot \sqrt{2}=14$

However, again the hypotenuse of the middle triangle is also the leg of the upper triangle. To find , the hypotenuse of the upper triangle, we simply repeat the process one last time.

$14\cdot \sqrt{2}=14\sqrt{2}$

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Example Question #1 : 30 60 90 Triangles

In a 30-60-90 triangle, the length of the side opposite the $30^\circ$ angle is . What is the length of the hypotenuse?

Possible Answers:

$10\sqrt{3}$

$5\sqrt{3}$

Correct answer:

Explanation:

By definition, the length of the hypotenuse is twice the length of the side opposite the $30^\circ$ angle.

Recall that the hypotenuse is the side opposite the $90^\circ$ angle.

Thus, using the equation below, where ss represents the short side (that opposite the $30^\circ$ angle) we get:

$H=2\cdot SS$

Plugging in our values for the short side we find the hypotenuse as follows:

$5\cdot2 = 10$

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Example Question #1 : 30 60 90 Triangles

A triangle has three angles , and such that B=2A and C=3A . The side opposite to measures units in length. How long is the side opposite of ?

Possible Answers:

$3\sqrt3$

$\sqrt3$

Correct answer:

Explanation:

A triangle with a 1:2:3 angle relation is a , , degree triangle. The side opposite the smallest angle of a triangle is the shortest side, of length . The side opposite the largest angle is the longest side, measuring twice the length of the shortest side for this triangle, units.

$\frac{1}{2}=\sin{30}$

$=\frac{opp}{hyp}=\frac{3}{h}$

Therefore, to make the above statement true h=6 .

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Example Question #2 : 30 60 90 Triangles

Triangle ABC is equilateral with a side length of .

What is the height of the triangle?

Possible Answers:

$\frac{s}{2}$

$\frac{\sqrt3}{2}s$

$\sqrt{s}$

$\frac{2s}{\sqrt3}$

Correct answer:

$\frac{\sqrt3}{2}s$

Explanation:

An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length,

$\sin(60)=\frac{height}{side}$

so..

$h=s*\sin(60)=s\frac{\sqrt3}{2}$

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Example Question #2 : 30 60 90 Triangles

In a 30-60-90 triangle, the side opposite the degree angle is . How long is the side opposite the degree angle?

Possible Answers:

$15\sqrt{2}$

$15\sqrt{3}$

Correct answer:

Explanation:

Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.

Therefore,

$h=15\cdot2$

h=30

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Trigonometry : Triangles

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : 45 45 90 Triangles

Example Question #1 : 45 45 90 Triangles

Example Question #2 : 45 45 90 Triangles

Example Question #3 : 45 45 90 Triangles

Example Question #1 : 45 45 90 Triangles

Example Question #2 : 45 45 90 Triangles

Example Question #1 : 30 60 90 Triangles

Example Question #1 : 30 60 90 Triangles

Example Question #2 : 30 60 90 Triangles

Example Question #2 : 30 60 90 Triangles

All Trigonometry Resources

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