Triangles - Trigonometry
Card 1 of 644
In a 
triangle, the side opposite the 
degree angle is 
. How long is the side opposite the 
degree angle?
In a
Tap to reveal answer
Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.
Therefore,
Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.
Therefore,
← Didn't Know|Knew It →
In a 30-60-90 triangle, the length of the side opposite the 
angle is 
. What is the length of the hypotenuse?
In a 30-60-90 triangle, the length of the side opposite the
Tap to reveal answer
By definition, the length of the hypotenuse is twice the length of the side opposite the 
angle.
Recall that the hypotenuse is the side opposite the 
angle.
Thus, using the equation below, where ss represents the short side (that opposite the 
angle) we get:
Plugging in our values for the short side we find the hypotenuse as follows:
By definition, the length of the hypotenuse is twice the length of the side opposite the
Recall that the hypotenuse is the side opposite the
Thus, using the equation below, where ss represents the short side (that opposite the
Plugging in our values for the short side we find the hypotenuse as follows:
← Didn't Know|Knew It →
A triangle has three angles 
, 
and 
such that 
and 
. The side opposite to 
measures 
units in length. How long is the side opposite of 
?
A triangle has three angles
Tap to reveal answer
A triangle with a 
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.
Therefore, to make the above statement true 
.
A triangle with a
Therefore, to make the above statement true
← Didn't Know|Knew It →
Triangle 
is equilateral with a side length of 
.
What is the height of the triangle?
Triangle
What is the height of the triangle?
Tap to reveal answer
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,
so..
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,
so..
← Didn't Know|Knew It →
What is the height of an equilateral triangle with side length 8?
What is the height of an equilateral triangle with side length 8?
Tap to reveal answer
The altitude of an equilateral triangle splits it into two 30-60-90 triangles. The height of the triangle is the longer leg of the 30-60-90 triangle. If the hypotenuse is 8, the longer leg is 
.
To double check the answer use the Pythagorean Thereom:
The altitude of an equilateral triangle splits it into two 30-60-90 triangles. The height of the triangle is the longer leg of the 30-60-90 triangle. If the hypotenuse is 8, the longer leg is
To double check the answer use the Pythagorean Thereom:
← Didn't Know|Knew It →
What is the ratio of the side opposite the 
angle to the hypotenuse?
What is the ratio of the side opposite the
Tap to reveal answer
Step 1: Locate the side that is opposite the 
side..
The shortest side is opposite the 
angle. Let's say that this side has length 
.
Step 2: Recall the ratio of the sides of a 
triangle:
From the shortest side, the ratio is 
.
is the hypotenuse, which is twice as big as the shortest side..
The ratio of the short side to the hypotenuse is 
Step 1: Locate the side that is opposite the
The shortest side is opposite the
Step 2: Recall the ratio of the sides of a
From the shortest side, the ratio is
The ratio of the short side to the hypotenuse is
← Didn't Know|Knew It →
It is known that the smallest side of a 30-60-90 triangle is 5.
Find 
.
It is known that the smallest side of a 30-60-90 triangle is 5.
Find
Tap to reveal answer
We know that in a 30-60=90 triangle, the smallest side corresponds to the side opposite the 30 degree angle.
Additionally, we know that the hypotenuse is 2 times the value of the smallest side, so in this case, that is 10.
The formula for
, so 
or 
.
We know that in a 30-60=90 triangle, the smallest side corresponds to the side opposite the 30 degree angle.
Additionally, we know that the hypotenuse is 2 times the value of the smallest side, so in this case, that is 10.
The formula for
← Didn't Know|Knew It →
It is known that for a 30-60-90 triangle,
.
Find the area of the triangle.
Note:
It is known that for a 30-60-90 triangle,
Find the area of the triangle.
Note:
Tap to reveal answer
First, we know that in a 30-60-90 triangle,
.
Also, the base is the smallest side times 
, so in our case it is 
.
The height is just the smallest side, 
.
Substituting these values into the formula given for area of a triangle, we obtain the answer 
.
First, we know that in a 30-60-90 triangle,
Also, the base is the smallest side times
The height is just the smallest side,
Substituting these values into the formula given for area of a triangle, we obtain the answer
← Didn't Know|Knew It →
In a 
triangle, if one leg is 
. What is the measure of the hypotenuse?
In a
Tap to reveal answer
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.
Or, you can remember the 45-45-90 identity, which states that the hypotenuse is 
times the leg.
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.
Or, you can remember the 45-45-90 identity, which states that the hypotenuse is
← Didn't Know|Knew It →
A triangle has three angles 
, 
, 
such that 
and 
together are as much as 
. What is the ratio of the longest side to the shortest?
A triangle has three angles
Tap to reveal answer
A triangle with the sum of two angles equaling the third is a 
triangle
A triangle with the sum of two angles equaling the third is a
← Didn't Know|Knew It →
Find the value of 
in the triangle below.
Find the value of
Tap to reveal answer
The first two things to recognize regarding our tirangle are 1) it is a right triangle and 2) it is an isosceles triangle. The two congruent sides tell us that the two non-right angles are also congruent, and a little quick math tells us that they each equal 45 degrees. This means our right triangle is not just any right triangle but a 45-45-90 triangle.
This is important because the sides of every 45-45-90 triangle follow the same ratio. The two legs are obviously always congruent to each other (being isosceles), but to find the hypotenuse, we simply have to multiply the length of a leg by 
.
Given this fact we would be in good shape if we had the length of a leg and needed the hypotenuse. But we have the hypotenuse and need the leg, which we means we need to work backwards going this way, we need to divide the length of the hypotenuse by 
. Therefore,
However, general practice in mathematics doesn't allow us to leave a square root in the denominator. We solve this problem by rationalizing the denominator, which is accomplished by multiplying the numerator and the denominator by 
.
This effectively eliminates the square root in the denominator and provides our answer.
The first two things to recognize regarding our tirangle are 1) it is a right triangle and 2) it is an isosceles triangle. The two congruent sides tell us that the two non-right angles are also congruent, and a little quick math tells us that they each equal 45 degrees. This means our right triangle is not just any right triangle but a 45-45-90 triangle.
This is important because the sides of every 45-45-90 triangle follow the same ratio. The two legs are obviously always congruent to each other (being isosceles), but to find the hypotenuse, we simply have to multiply the length of a leg by
Given this fact we would be in good shape if we had the length of a leg and needed the hypotenuse. But we have the hypotenuse and need the leg, which we means we need to work backwards going this way, we need to divide the length of the hypotenuse by
However, general practice in mathematics doesn't allow us to leave a square root in the denominator. We solve this problem by rationalizing the denominator, which is accomplished by multiplying the numerator and the denominator by
This effectively eliminates the square root in the denominator and provides our answer.
← Didn't Know|Knew It →
The following figure was made by beginning with a square. The midpoints of the four sides of the square were then joined to form another square. The process was repeated to form a third square and finally once more to form the fourth and smallest square in the middle, which has a side length of 
. Find the value of 
.
The following figure was made by beginning with a square. The midpoints of the four sides of the square were then joined to form another square. The process was repeated to form a third square and finally once more to form the fourth and smallest square in the middle, which has a side length of
Tap to reveal answer
We begin by realizing that the midpoints of the sides of our outer square divide each side in half. Furthermore, the sides of our second square connecting these midpoints form four right triangles in each corner of our largest square.
But these right triangles are special right triangles. They are 45-45-90 triangles, which means we can find the hypotenuse (and thus the side of our second square) by multiplying the length of the leg by 
. Therefore the length of a side of our second square is 
.
We now repeat the process, beginning by forming four new 45-45-90 triangles
To find the hypotenuse of each of these triangles (and thus the side length of our third square), we simply multiply by 
again.
We then repeat the process one final time, multiplying by 
again.
Our final hypotenuse and thus the side of our innermost square is 
.
We begin by realizing that the midpoints of the sides of our outer square divide each side in half. Furthermore, the sides of our second square connecting these midpoints form four right triangles in each corner of our largest square.
But these right triangles are special right triangles. They are 45-45-90 triangles, which means we can find the hypotenuse (and thus the side of our second square) by multiplying the length of the leg by
We now repeat the process, beginning by forming four new 45-45-90 triangles
To find the hypotenuse of each of these triangles (and thus the side length of our third square), we simply multiply by
We then repeat the process one final time, multiplying by
Our final hypotenuse and thus the side of our innermost square is
← Didn't Know|Knew It →
Find the value of 
Find the value of
Tap to reveal answer
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 
. Therefore, the hypotenuse of our bottom triangle is
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.
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.
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
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.
However, again the hypotenuse of the middle triangle is also the leg of the upper triangle. To find
← Didn't Know|Knew It →
In a 45-45-90 triangle, if the hypotenuse is 10, what is the perimeter of the triangle?
In a 45-45-90 triangle, if the hypotenuse is 10, what is the perimeter of the triangle?
Tap to reveal answer
Write the Pythagorean Theorem.
In a 45-45-90 triangle, the length of the legs are equal, which indicates that:
Rewrite the formula and substitute the known sides.
The lengths of the triangle are: 
Sum the three lengths for the perimeter.
Write the Pythagorean Theorem.
In a 45-45-90 triangle, the length of the legs are equal, which indicates that:
Rewrite the formula and substitute the known sides.
The lengths of the triangle are:
Sum the three lengths for the perimeter.
← Didn't Know|Knew It →
One side of a 
-
-
triangle has a length of 3. Which cannot be the length of one of the other sides?
One side of a
Tap to reveal answer
If 3 is one of the legs, then the hypotenuse is 
.
If 3 is the hypotenuse, then the legs are 
or equivalently 
If 3 is one of the legs, then the hypotenuse is
If 3 is the hypotenuse, then the legs are
← Didn't Know|Knew It →
The hypotenuse of a 
-
-
triangle is 4. What is the length of each of the legs?
The hypotenuse of a
Tap to reveal answer
Divide the length of the hypotenuse by 
to get the length of the legs:
Divide the length of the hypotenuse by
← Didn't Know|Knew It →
The perimeter of a square is 56 feet. What is the length of the diagonal in feet?
The perimeter of a square is 56 feet. What is the length of the diagonal in feet?
Tap to reveal answer
The perimeter of a square can be found using the formula 
, where P is the perimeter and s is the length of the side of the square.
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 
, so the diagonal of this square is 
.
The perimeter of a square can be found using the formula
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
← Didn't Know|Knew It →
In right triangle ABC, where angle A measures 90 degrees, side AB measures 15 and side AC measures 36, what is the length of side BC?
In right triangle ABC, where angle A measures 90 degrees, side AB measures 15 and side AC measures 36, what is the length of side BC?
Tap to reveal answer
This triangle can exist. Since 
is a right angle, we can use the Pythagorean Theorem, where 
is the hypoteneuse:
This triangle can exist. Since
← Didn't Know|Knew It →
While waiting for your sister to finish her bungee jump, you decide to figure out how tall the platform she is jumping off is. You are standing 
feet from the base of the platform, and the angle of elevation from your position to the top of the platform is 
degrees. How many feet tall is the platform?
While waiting for your sister to finish her bungee jump, you decide to figure out how tall the platform she is jumping off is. You are standing
Tap to reveal answer
You can draw the following right triangle using the information given by the question:
Since you want to find the height of the platform, you will need to use tangent.
Make sure to round to 
places after the decimal.
You can draw the following right triangle using the information given by the question:
Since you want to find the height of the platform, you will need to use tangent.
Make sure to round to
← Didn't Know|Knew It →
When the angle of elevation of the sun is 
degrees, a flagpole casts a shadow that is 
feet long. In feet, how tall is the flagpole?
When the angle of elevation of the sun is
Tap to reveal answer
You can draw the following right triangle from the information given by the question.
In order to find the height of the flagpole, you will need to use tangent.
Make sure to round to 
places after the decimal.
The flagpole is 
feet tall.
You can draw the following right triangle from the information given by the question.
In order to find the height of the flagpole, you will need to use tangent.
Make sure to round to
The flagpole is
← Didn't Know|Knew It →