We know that the perimeter is 36 inches, and one side is 12. This means, the sum of the lengths of the other two sides are 24. The ratio between the two sides is 3:5, giving a total of 8 parts. We divide the remaining length, 24 inches, by 8 giving us 3. This means each part is 3. We multiply this by the ratio and get 9:15, meaning the longest side is 15 inches.

What is the value of  in the triangle above? Round to the nearest hundredth.

Begin by filling in the missing angle for your triangle. Since a triangle has a total of  degrees, you know that the missing angle is:

Draw out the figure:

Now, to solve this, you will need some trigonometry. Use the Law of Sines to calculate the value:

Solving for , you get:

Rounding, this is .

Begin by filling in the missing angle for your triangle. Since a triangle has a total of  degrees, you know that the missing angle is:

Draw out the figure:

Now, to solve this, you will need some trigonometry. Use the Law of Sines to calculate the value:

Solving for , you get:

Rounding, this is .

Begin by filling in the missing angle for your triangle. Since a triangle has a total of  degrees, you know that the missing angle is:

Draw out the figure:

This problem becomes incredibly easy! This is an isosceles triangle. Therefore, you know that  is , because it is "across" from a  degree angle—which matches the other  degree angle!

Recall that an isosceles right triangle is a  triangle. That means that it looks like this:

This makes calculating the area very easy! Recall, the area of a triangle is defined as:

However, since  for our triangle, we know:

Now, we know that . Therefore, we can write:

Solving for , we get:

This is the length of the height of the triangle for the side that is not the hypotenuse.

Based on the information given, you know that your triangle looks as follows:

This is a  triangle. Recall your standard  triangle:

You can set up the following ratio between these two figures:

Now, the area of the triangle will merely be  (since both the base and the height are ). For your data, this is:

To solve simply realize the hypotenuse of one of these triangles is of the form where s is side length. Thus, our answer is .

Recall that an isosceles right triangle is a  triangle. That means that it looks like this:

This makes calculating the area very easy! Recall, the area of a triangle is defined as:

However, since  for our triangle, we know:

Now, we know that . Therefore, we can write:

Solving for , we get:

However, be careful! Notice what the question asks: "What is its height that is correlative and perpendicular to this triangle's hypotenuse?" First, let's find the hypotenuse of the triangle. Recall your standard  triangle:

Since one of your sides is , your hypotenuse is .

Okay, what you are actually looking for is  in the following figure:

Therefore, since you know the area, you can say:

Solving, you get: . 

Now, this is really your standard  triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be  degrees, because the triangle is isosceles.

Based on the description of your triangle, you can draw the following figure:

This is derived from your reference triangle for the  triangle:

For our triangle, we could call one of the legs . We know, then:

Thus, .

The area of your triangle is:

For your data, this is:

Now, this is really your standard  triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be  degrees because the triangle is isosceles.

Based on the description of your triangle, you can draw the following figure:

This is derived from your reference triangle for the  triangle:

For our triangle, we could call one of the legs . We know, then:

Thus, .

The area of your triangle is:

For your data, this is: