ACT Math : How to find the perimeter of an acute / obtuse isosceles triangle
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Example Questions
Example Question #4 : Acute / Obtuse Isosceles Triangles
The height of an isosceles triangle, dropped from the vertex to its base, is one fourth the length of the base. If the area of this triangle is 
Based on the description of this question, you can draw your triangle as such. We can do this thanks to the nature of an isosceles triangle:
Now, you know that the area of a triangle is defined as:
So, for our data, we can say:
Solving for 
Thus, 
Now, for our little triangle on the right, we can draw:
Using the Pythagorean Theorem, we know that the other side is:
This can be simplified to:
Now, we know that this side is the "equal" side of the isosceles triangle. Therefore, we can know that the total perimeter is:
Example Question #5 : Acute / Obtuse Isosceles Triangles
The base of an isosceles triangle is five times the length of its correlative height. If the area of this triangle is 
Based on the description of this question, you can draw your triangle as such. We can do this thanks to the nature of an isosceles triangle:
Now, you know that the area of a triangle is defined as:
So, for our data, we can say:
Solving for 
Thus, 
Now, for our little triangle on the right, we can draw:
Using the Pythagorean Theorem, we know that the other side is:
This can be simplified to:
Now, we know that this side is the "equal" side of the isosceles triangle. Therefore, we can know that the total perimeter is:
Example Question #6 : Acute / Obtuse Isosceles Triangles
What is the area of an isosceles triangle with a vertex of 
Based on the description of your triangle, you can draw the following figure:
You can do this because you know:
- The two equivalent sides are given.
- Since a triangle is
degrees, you have only
or
degrees left for the two angles of equal size. Therefore, those two angles must be
degrees and
degrees.
Now, based on the properties of an isosceles triangle, you can draw the following as well:
Based on your standard reference 
Therefore, 
This means that 
Now, the area of the triangle is:
Example Question #1 : Acute / Obtuse Isosceles Triangles
What is the perimeter of an isosceles triangle with a vertex of 
Based on the description of your triangle, you can draw the following figure:
You can do this because you know:
- The two equivalent sides are given.
- Since a triangle is
degrees, you have only
or
degrees left for the two angles of equal size. Therefore, those two angles must be
degrees and
degrees.
Now, based on the properties of an isosceles triangle, you can draw the following as well:
Based on your standard reference 
Therefore, 
This means that 
Therefore, the perimeter of the triangle is:
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