ACT Math : How to find the perimeter of an acute / obtuse isosceles triangle

Study concepts, example questions & explanations for ACT Math

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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 , what is its perimeter?

Possible Answers:

Correct answer:

Explanation:

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:

_tri41

Now, you know that the area of a triangle is defined as:

So, for our data, we can say:

Solving for , we get:

Thus, .

Now, for our little triangle on the right, we can draw:

_tri53

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 , what is its perimeter?

Possible Answers:

Correct answer:

Explanation:

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:

_tri51

Now, you know that the area of a triangle is defined as:

So, for our data, we can say:

Solving for , we get:

Thus, .

Now, for our little triangle on the right, we can draw:

_tri54

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  degrees and two sides equal to  units?

Possible Answers:

Correct answer:

Explanation:

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

_tri91

You can do this because you know:

  1. The two equivalent sides are given.
  2. 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:

_tri92

Based on your standard reference  triangle, you know:

Therefore,  is .

This means that  is , and the total base of the triangle is .

Now, the area of the triangle is:

 or  

Example Question #7 : Acute / Obtuse Isosceles Triangles

What is the perimeter of an isosceles triangle with a vertex of  degrees and two sides equal to 

Possible Answers:

Correct answer:

Explanation:

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

_tri81

You can do this because you know:

  1. The two equivalent sides are given.
  2. 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:

_tri82

Based on your standard reference  triangle, you know:

Therefore,  is .

This means that  is  and the total base of the triangle is .

Therefore, the perimeter of the triangle is:

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