Question

solve for b given that the height of the triangle is 8 centimeters.

Explanation:

Step1: Identify the triangle type

The triangle is isosceles, and the height splits the base into two equal parts. So the base of each right triangle formed is $\frac{4}{2} = 2$ cm.

Step2: Apply Pythagorean theorem

For a right triangle, $a^2 + b^2 = c^2$, where $a = 2$ (base), $b = 8$ (height), and $c = b$ (hypotenuse we need to find). Wait, correction: Let the hypotenuse be $b$, one leg is height $8$, the other leg is half of the base, which is $\frac{4}{2}=2$. So using Pythagorean theorem: $2^2 + 8^2 = b^2$.

Step3: Calculate $b^2$

$2^2 + 8^2 = 4 + 64 = 68$? Wait, no, wait the base of the whole triangle is 4, so when we draw the height, it splits the base into two segments of length $\frac{4}{2}=2$. So the right triangle has legs 2 and 8, hypotenuse $b$. So $b^2 = 2^2 + 8^2 = 4 + 64 = 68$? Wait, no, wait maybe I misread the base. Wait the base of the whole triangle is 4? Wait the diagram shows the base as 4? Wait the bottom side is 4, so when we draw the height (8) from the top vertex to the base, it splits the base into two equal parts, each of length 2. So the right triangle has legs 2 (horizontal) and 8 (vertical), and hypotenuse $b$. So by Pythagoras: $b = \sqrt{2^2 + 8^2} = \sqrt{4 + 64} = \sqrt{68} = 2\sqrt{17} \approx 8.246$? Wait, no, wait maybe the base is 4, but maybe I made a mistake. Wait the problem says "the height of the triangle is 8 centimeters". Wait the diagram: the two sides are $b$, the height is 8, and the base is 4. So when we drop the height, it's perpendicular to the base, so we have two right triangles, each with base 2, height 8, hypotenuse $b$. So Pythagorean theorem: $b^2 = 2^2 + 8^2 = 4 + 64 = 68$, so $b = \sqrt{68} = 2\sqrt{17} \approx 8.246$? Wait, but maybe the base is 4, but maybe I misread. Wait no, wait the base of the whole triangle is 4, so half is 2. So yes, the right triangle has legs 2 and 8, hypotenuse $b$. So $b = \sqrt{2^2 + 8^2} = \sqrt{4 + 64} = \sqrt{68} = 2\sqrt{17} \approx 8.25$ (rounded to two decimal places) or in exact form $2\sqrt{17}$. Wait, but maybe I made a mistake in the base. Wait the diagram: the bottom side is 4, so the length of the base is 4, so when we draw the height, it's 8, so the two right triangles have base 2, height 8, hypotenuse $b$. So that's correct.

Wait, but let's recheck. The Pythagorean theorem: in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So here, one side is 2 (half of 4), one side is 8 (height), so hypotenuse $b$: $b^2 = 2^2 + 8^2 = 4 + 64 = 68$, so $b = \sqrt{68} = 2\sqrt{17} \approx 8.25$ cm.

Answer:

$b = 2\sqrt{17}$ centimeters (or approximately $8.25$ centimeters)