Question

If the length of the altitudes of a triangle is 3 cm, 4 cm, and 5 cm, then find the inradius of the triangle.

a. 1 cm
b. 2 cm
c. 3 cm
d. 4 cm

Answer 1

The inradius of the triangle is 1 cm.

If the lengths of the altitudes of a triangle are 3 cm, 4 cm, and 5 cm, and you've deduced that it is a right-angled triangle using the Pythagorean theorem, then the triangle is indeed a special case known as a Pythagorean triple.

The lengths 3 cm, 4 cm, and 5 cm form a Pythagorean triple because they satisfy the Pythagorean theorem:

`\[ a^2 + b^2 = c^2 \]`

where
`\( a = 3 \), \( b = 4 \), and \( c = 5 \)`.

In a right-angled triangle, the inradius
`(\( r \))` can be found using the formula:

`\[ r = (a + b - c)/(2) \]`

Substitute the values:

`\[ r = (3 + 4 - 5)/(2) = (2)/(2) = 1 \]`

Therefore, the inradius of the triangle is 1 cm.