An acute isosceles triangle is a triangle that has two sides of equal length and whose interior angles are acute. The isosceles triangle is characterized by having two sides with the same length and two angles with the same measure. On the other hand, an acute triangle is characterized by having only acute internal angles, that is, less than 90 degrees.

Therefore, the isosceles acute triangle is a triangle that meets the conditions of both an isosceles triangle and an acute triangle.

## Characteristics of acute isosceles triangles

Acute isosceles triangles have the following characteristics:

- Two sides have the same length, that is, two sides are congruent.
- The side that is uneven to the other sides is called the base of the triangle.
- All interior angles are acute, that is, they measure less than 90 degrees.
- The angles opposite the two equal sides have the same measure, that is, the angles of the bases are congruent.
- The third angle, which is different from the base angles, is called the apex angle.
- The line perpendicular to the base and connecting to the apex angle is the height.
- The height bisects the base of the triangle into two equal parts.
- The height also bisects the apex angle into two equal angles.
- The height divides the triangle into two congruent right triangles.

## Commonly used isosceles triangle formulas

The perimeter, area, and height formulas are the most used and can help us solve problems of acute isosceles triangles.

### Perimeter of isosceles triangles

The perimeter of any figure is equal to the sum of the lengths of all its sides. In isosceles triangles, we can modify the perimeter formula to define that two sides are equal:

$latex p=b+2a$ |

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

### Isosceles triangle area

We can calculate the area of any triangle by multiplying the length of its base by the length of its height and dividing by 2:

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

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

### Height of isosceles triangles

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

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

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

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## Examples of isosceles triangle problems

### EXAMPLE 1

- An isosceles triangle has a base of length 11 m and congruent sides of length 12 m. What is its perimeter?

**Solution:** We can recognize the following:

- Base, $latex b=11$ m
- Sides, $latex a=12$ m

We substitute those values in the perimeter formula:

$latex p=b+2a$

$latex p=11+2(12)$

$latex p=11+24$

$latex p=35$

The perimeter is 35 m.

### EXAMPLE 2

- What is the area of a triangle that has a base of 10 m and a height of 14 m?

**Solution:** We have the following information:

- Base, $latex b=10$ m
- Height, $latex h=14$ m

We use the area formula with these values:

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

$latex A= \frac{1}{2}(10)(14)$

$latex A=70$

The area is 70 m².

### EXAMPLE 3

- An isosceles triangle has a base of length 10 m and congruent sides of length 12 m. What is its height?

**Solution:** We can use the following values:

- Base, $latex b=10$ m
- Sides, $latex a=12$ m

By substituting these values in the formula for the area, we have:

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

$latex h= \sqrt{{{12}^2}- \frac{{{10}^2}}{4}}$

$latex h= \sqrt{144- \frac{100}{4}}$

$latex h= \sqrt{144-25}$

$latex h= \sqrt{119}$

$latex h=10.9$

The height of the triangle is 10.9 m.

## Isosceles triangles – Practice problems

## See also

Interested in learning more about isosceles triangles? Take a look at these pages:

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