Question

Entrance of a museum is the shape of an isosceles triangle with height of 12 and slop of 16 what is the length

Answer 1

To find the length of the base of the isosceles triangle given the height and the lengths of the two equal sides (the slopes), we can use the properties of right triangles.

The height divides the isosceles triangle into two right triangles. Each of these right triangles has:

- One leg as the height (12 units),
- The other leg as half of the base (let's call it \( \frac{b}{2} \)),
- The hypotenuse as the length of the sides (16 units).

Using the Pythagorean theorem, we can set up the following equation:

\[
\left(\frac{b}{2}\right)^2 + 12^2 = 16^2
\]

Calculating \( 12^2 \) and \( 16^2 \):

\[
\left(\frac{b}{2}\right)^2 + 144 = 256
\]

Subtracting 144 from both sides:

\[
\left(\frac{b}{2}\right)^2 = 256 - 144
\]

\[
\left(\frac{b}{2}\right)^2 = 112
\]

Taking the square root of both sides:

\[
\frac{b}{2} = \sqrt{112}
\]

Calculating \( \sqrt{112} \):

\[
\sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7}
\]

Thus,

\[
\frac{b}{2} = 4\sqrt{7}
\]

Now, multiplying both sides by 2 to find \( b \):

\[
b = 8\sqrt{7}
\]

Therefore, the length of the base of the isosceles triangle is \( 8\sqrt{7} \) units. If you need a decimal approximation:

\[
\sqrt{7} \approx 2.64575 \Rightarrow 8\sqrt{7} \approx 8 \times 2.64575 \approx 21.166
\]

So the base length is approximately **21.17 units**.