Right Triangles - Trigonometry

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:

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 .

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..

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:

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

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 .

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 .

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.

A triangle with the sum of two angles equaling the third is a triangle

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.

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 .

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.

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.

If 3 is one of the legs, then the hypotenuse is .

If 3 is the hypotenuse, then the legs are or equivalently

Divide the length of the hypotenuse by to get the length of the legs:

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 .

This triangle can exist. Since is a right angle, we can use the Pythagorean Theorem, where is the hypoteneuse:

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 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.