Question

The height of an isosceles triangle is 6cm. Find the third side if the length of the two equal sides is 10cm

Answer 1

Explanation:

In an isosceles triangle, the two equal sides are opposite to the two equal angles, and the third side is opposite to the base angle.

Let's draw the triangle and label the height h, the length of the equal sides a, and the length of the base b:

/|\

/ | \

a / | \ a

/ |h \

/____|____\

b

We know that the length of the equal sides is 10 cm, so a = 10 cm.

We also know that the height is 6 cm, so h = 6 cm.

To find the length of the base b, we can use the Pythagorean theorem, which relates the sides of a right triangle:

b^2 = a^2 - h^2

Substituting the given values, we get:

b^2 = 10^2 - 6^2

b^2 = 100 - 36

b^2 = 64

Taking the square root of both sides, we get:

b = 8

Therefore, the length of the third side (the base) is 8 cm.

Answer 2

To solve this problem, we can use the Pythagorean theorem and the fact that the altitude of an isosceles triangle bisects the base. Let's call the third side of the triangle "x".

First, we can draw the isosceles triangle and label its height and two equal sides:

```

/|\

/ | \

6cm / | \ 6cm

/ |h \

/____|____\

10cm

```

We can see that the altitude of the triangle divides the base into two equal segments of length 5cm each. We can use the Pythagorean theorem to find the length of the third side:

```

x^2 = h^2 + 5^2 (by Pythagorean theorem)

x^2 = 6^2 + 5^2

x^2 = 61

x = sqrt(61)

```

Therefore, the length of the third side is approximately 7.81cm (rounded to two decimal places).