Intermediate Geometry : How to find the length of the side of of an acute / obtuse isosceles triangle

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #1 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

Find the perimeter of the triangle below.

11

Possible Answers:

Correct answer:

Explanation:

Use the Pythagorean Theorem to find the base of the right triangle.

Now, because two of the angles in this triangle are the same, this is an isosceles triangle. In an isosceles triangle, the sides that are directly across from the congruent angles are also congruent.

In addition, the height in an isosceles triangle will always cut the 3rd side in half. With this information, fill out the triangle as shown below:

11a

To find the perimeter, add up all the sides.

Example Question #1 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

Find the perimeter of the triangle below.

12

Possible Answers:

Correct answer:

Explanation:

Use the Pythagorean Theorem to find the base of the right triangle.

Now, because two of the angles in this triangle are the same, this is an isosceles triangle. In an isosceles triangle, the sides that are directly across from the congruent angles are also congruent.

In addition, the height in an isosceles triangle will always cut the 3rd side in half. With this information, fill out the triangle as shown below:

16

To find the perimeter, add up all the sides.

Example Question #52 : Acute / Obtuse Isosceles Triangles

A triangle has sides of lengths 14, 18, and 20. Is the triangle scalene or isosceles?

Possible Answers:

Isosceles

Scalene

Correct answer:

Scalene

Explanation:

A triangle with three sides of different length is, by definition, scalene.

Example Question #2 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

Given: Regular Pentagon  with center . Construct segments  and  to form  .

True or false:  is an isosceles triangle.

Possible Answers:

True 

False

Correct answer:

True 

Explanation:

Below is regular Pentagon  with center , a segment drawn from  to each vertex - that is, each of its radii drawn.

Pentagon a

 

By symmetry, all of the radii of a regular pentagon are congruent - specifically, . This triangle has at least two congruent sides, so it is isosceles.

Example Question #3 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

If a triangle has side lengths of  and , which of the following can be a length of the third side?

Possible Answers:

Correct answer:

Explanation:

13

The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. The relationship can be represented by the following inequalities:

The side length of  is the only choice that fits this criteria:

Example Question #5 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

If a triangle has side lengths of  and , which of the following could be the length of the third side?

Possible Answers:

Correct answer:

Explanation:

13

The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. The relationship can be represented by the following inequalities:

The side length of  is the only choice that fits this criteria:

Example Question #6 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

Isosceles

Figure NOT drawn to scale.

Refer to the above diagram.

True or false: .

Possible Answers:

True

False

Correct answer:

True

Explanation:

The sum of the measures of the interior angles of a triangle is , so 

Substitute the given two angle measures and solve for :

Subtract  from both sides:

, so, by the Converse of the Isosceles Triangle Theorem, their opposite sides are also congruent - that is,

Example Question #7 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

Isosceles

Refer to the above triangle. By what statement does it follow that ?

Possible Answers:

The Angle-Angle Postulate

The Angle-Side-Angle Postulate

The Converse of the Pythagorean Theorem

The Hinge Theorem

The Converse of the Isosceles Triangle Theorem

Correct answer:

The Converse of the Isosceles Triangle Theorem

Explanation:

We are given that, in , two angles are congruent; specifically, . It is a consequence of the Converse of the Isosceles Triangle Theorem that the sides opposite the angles are also congruent - that is, ..

Example Question #8 : How To Find The Length Of The Side Of Of An Acute / Obtuse Isosceles Triangle

Given  such that , which of the following statements is true?

Possible Answers:

It cannot be determined from the information given.

Correct answer:

Explanation:

Having three sides of different lengths, this triangle is scalene. In any scalene triangle, the angle with greatest measure is opposite the longest side, and the angle with least measure is opposite the shortest side. Therefore, since , their opposite angles would be in order from greatest to least measure - that is, .

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