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Isosceles Acute Triangle

An isosceles acute triangle is a triangle in which all three angles are less than 90 degrees and at least two of its angles are equal in measurement. One example of isosceles acute triangle angles is 50°, 50°, and 80°.

1. Isosceles Acute Triangle Definition
2. Properties of Isosceles Acute Triangle
3. Isosceles Acute Triangle Formulas
4. FAQs on Isosceles Acute Triangle

Isosceles Acute Triangle Definition

In geometry, an isosceles acute triangle can be considered as the triangle that contains the properties of both an isosceles triangle and an acute triangle. Let us recap the meaning of isosceles triangles and acute triangles.

  • An isosceles triangle is one in which any two angles of the triangle are equal in measurement. And the two sides opposite to those equal angles are also equal in length.
  • An acute triangle is one in which all three angles measure less than 90 degrees.

Look at the picture of an acute isosceles triangle given below to understand how it appears.

Properties of Isosceles Acute Triangle

It is easy to identify an isosceles acute triangle if we know its properties. The properties of the isosceles acute triangle are listed below:

  • Two angles and two sides opposite to those angles are equal.
  • All three angles are less than 90 degrees (acute angles).
  • The sum of all the interior angles is 180 degrees.

Isosceles Acute Triangle Formulas

The formula of an isosceles acute triangle is useful to find the area and perimeter of the triangle. There are two possible formulae that can be used to find the area of an isosceles acute triangle based on what information is given to us.

  • If the length of base and height of the triangle is given, then area = [1/2 × base × height] square units.
  • If the length of all three sides are given, then area = \((s-a) \sqrt{s(s-b)}\) square units, where s = perimeter/2 = (2a + b)/2. This formula is derived by using the Heron’s formula. Let’s see how.

Applying Heron’s formula to find the area of a triangle, we have, \(\sqrt{(s)(s-a)(s-b)(s-c)}\), where s is the semi-perimeter and a, b, and c are the sides of the triangle. But in the case of an isosceles acute triangle, two of the sides are equal. So let us assume a = c. By substituting the value of ‘c’ in the above formula, we get, \(\sqrt{(s)(s-a)(s-a)(s-b)}\).

⇒ \(\sqrt{(s)(s-a)^{2} (s-b)}\)

⇒ \((s-a)\sqrt{(s)(s-b)}\)

where a and b are the sides of the triangle and s is the semi-perimeter, which is (a + a + b)/2 or (2a+b)/2. Look at the image given below showing isosceles acute triangle formulas for finding area and perimeter.

To find the isosceles acute triangle perimeter, we just have to add the length of all three sides. So, the perimeter of an isosceles acute triangle = (2a + b) units, where a and b are the sides of the triangle.

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Isosceles Acute Triangle Examples

  1. Example 1: Find the area of an isosceles acute triangle if the base is 12 units and the height is 8 units.

    Solution: Given, base (b) = 12 units and height (h) = 8 units. The formula to calculate the area is 1/2 × b × h square units. By substituting the values, we get area = 1/2 × 12 × 8.

    ⇒ Area = 6 × 8 = 48

    Therefore, the area of the given isosceles acute triangle is 48 square units.

  2. Example 2: What will be the perimeter of an isosceles acute triangle if the sides are of lengths 5 units, 8 units, and 8 units?

    Solution: The perimeter is the sum of all the sides of a shape. Here, the given three side lengths are 5 units, 8 units, and 8 units. So, the perimeter = 5 + 8 + 8 units, which is 21 units. Therefore, the perimeter of the given isosceles acute triangle is 21 units.

  3. Example 3: If two of the angles of a triangle are given as 70° and 40°, find the value of the missing angle. Also, state whether these angles can form an isosceles acute triangle or not?

    Solution: The given angles are 70° and 40°. We know that the sum of all the angles of a triangle is always 180 degrees. So, let us assume that the value of the third angle be x°.

    ⇒ 70° + 40° + x° = 180°

    ⇒ 110° + x° = 180°

    ⇒ x = 180 – 110 = 70

    Therefore, the value of the missing angle is 70°. Since two angles are of the same measurements (70°) and all three angles are acute, we can form an isosceles acute triangle with the given angles.

Practice Questions on Isosceles Acute Triangle

FAQs on Isosceles Acute Triangle

What is an Isosceles Acute Triangle?

An isosceles acute triangle is a triangle that contains the properties of both the acute triangle and isosceles triangle. It comes in the category of both acute triangles and isosceles triangles. At least two of its angles are equal in measurement and all three angles are acute angles.

What is the Area of an Isosceles Acute Triangle?

The area of an acute isosceles triangle can be calculated by using the formula: Area = 1/2 × base × height square units. The base is the side opposite to the vertex from where the height is drawn or measured. It is usually the unequal side of the isosceles acute triangle.

What are the Properties of an Isosceles Acute Triangle?

The properties of an isosceles acute triangle are listed below:

  1. All three angles are acute (less than 90 degrees).
  2. Have at least two equal sides and two equal angles.

How do you Draw an Acute Isosceles Triangle?

To draw an isosceles acute triangle, the first step is to draw a line segment horizontally which will be the base of the triangle. Now, draw two angles of equal measurements (each should be less than 90 degrees) on both the ends of the line segment. Join the arms of the angles. In this way, we will get an acute isosceles triangle.

What does an Isosceles Acute Triangle Look Like?

An isosceles acute triangle looks like an acute triangle with two equal sides and two equal angles less than 90 degrees.

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