Isosceles obtuse triangles are triangles that have two sides of the same length and an angle greater than 90 degrees. These triangles meet the conditions for an isosceles triangle and an obtuse triangle at the same time. Recall that an isosceles triangle has two sides that have the same length and two angles that have the same measure.

On the other hand, an obtuse triangle is characterized by having an internal angle that measures more than 90 degrees.

## Characteristics of obtuse isosceles triangles

Obtuse isosceles triangles have the following characteristics:

- Two sides of the triangle are congruent (they are equal in length).
- The side that does not have the same length is called the base of the triangle.
- An internal angle of the triangle is obtuse, that is, it has more than 90 degrees.
- Angles on opposite equal sides are also equal and acute.
- The angle different from the other two is called the apex angle.
- The apex angle is the obtuse angle.
- The height is the line perpendicular to the base and joining the apex angle.
- The height divides the base into two equal parts, as well as the apex angle.
- The height divides the triangle into two congruent right triangles.

## Important isosceles triangle formulas

With the following formulas, we can solve a large number of problems related to isosceles triangles.

### Isosceles triangle perimeter formula

The perimeter of any geometric figure is calculated by adding the lengths of the sides of the figure. The formula for the perimeter of isosceles triangles considers the fact that two sides of the triangle are equal:

$latex p=b+2a$ |

where *b* is the length of the base and *a* is the length of the congruent sides.

### Isosceles triangle area formula

The formula to calculate the area of any triangle is as follows:

$latex A= \frac{1}{2} \times b \times h$ |

where *b* represents the length of the base and *h* represents the length of the height.

### Formula for the height of isosceles triangles

The height formula is derived from the Pythagorean theorem, where we use the lengths of the base and the congruent sides:

$latex h= \sqrt{{{a}^2}- \frac{{{b}^2}}{4}}$ |

where *a* is the length of the congruent sides and *b* is the length of the base.

## Examples with answers of isosceles triangle problems

### EXAMPLE 1

- An isosceles triangle has a base with a length of 15 m and its congruent sides are 12 m. What is its perimeter?

**Solution:** From the question, we get the following values:

- Base, $latex b=15$ m
- Sides, $latex a=12$ m

Using the perimeter formula, we have:

$latex p=b+2a$

$latex p=15+2(12)$

$latex p=15+24$

$latex p=39$

The perimeter is 39 m.

### EXAMPLE 2

- What is the area of a triangle that has a base of 20 m and a height of 15 m?

**Solution:** We recognize the following values:

- Base, $latex b=20$ m
- Height, $latex h=15$ m

We use the area formula with these values:

$latex A= \frac{1}{2}bh$

$latex A= \frac{1}{2}(20)(15)$

$latex A=150$

The area is 150 m².

### EXAMPLE 3

- An isosceles triangle has a base of length 16 m and congruent sides of length 10 m. What is its height?

**Solution:** We have the following values:

- Base, $latex b=16$ m
- Sides, $latex a=10$ m

We use the height formula with these values:

$latex h= \sqrt{{{a}^2}- \frac{{{b}^2}}{4}}$

$latex h= \sqrt{{{10}^2}- \frac{{{16}^2}}{4}}$

$latex h= \sqrt{100- \frac{256}{4}}$

$latex h= \sqrt{100-64}$

$latex h= \sqrt{36}$

$latex h=6$

The height of the triangle is 6 m.

## Isosceles triangles – Practice problems

## See also

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