Answer:
![h = 5\ cm](https://tex.z-dn.net/?f=h%20%3D%205%5C%20cm)
Step-by-step explanation:
Let's call B at the base of the triangle and call h at the height of the triangle. Then we know that:
The height of a triangle is 5 cm shorter than its base. This means that:
.
The area of the triangle is 25 cm²
By definition the area of a triangle is:
![A = 0.5Bh](https://tex.z-dn.net/?f=A%20%3D%200.5Bh)
For this triangle we know that
and
. We substitute these values in the equation and solve for B.
![25 = 0.5B (B-5)](https://tex.z-dn.net/?f=25%20%3D%200.5B%20%28B-5%29)
![0.5B ^ 2-\frac{5}{2}B-25 = 0](https://tex.z-dn.net/?f=0.5B%20%5E%202-%5Cfrac%7B5%7D%7B2%7DB-25%20%3D%200)
Now we use the quadratic formula to solve the equation.
For an equation of the form
the quadratic formula is:
![B=\frac{-b\±\sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=B%3D%5Cfrac%7B-b%5C%C2%B1%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
In this case note that: ![a=0.5,\ \ b=-\frac{5}{2}\ \ c=-25](https://tex.z-dn.net/?f=a%3D0.5%2C%5C%20%5C%20b%3D-%5Cfrac%7B5%7D%7B2%7D%5C%20%5C%20c%3D-25)
Then:
![B=\frac{-(-\frac{5}{2})\±\sqrt{(-\frac{5}{2})^2-4(0.5)(-25)}}{2(0.5)}](https://tex.z-dn.net/?f=B%3D%5Cfrac%7B-%28-%5Cfrac%7B5%7D%7B2%7D%29%5C%C2%B1%5Csqrt%7B%28-%5Cfrac%7B5%7D%7B2%7D%29%5E2-4%280.5%29%28-25%29%7D%7D%7B2%280.5%29%7D)
![B=\frac{\frac{5}{2}\±\sqrt{\frac{25}{4}+50}}{1}](https://tex.z-dn.net/?f=B%3D%5Cfrac%7B%5Cfrac%7B5%7D%7B2%7D%5C%C2%B1%5Csqrt%7B%5Cfrac%7B25%7D%7B4%7D%2B50%7D%7D%7B1%7D)
![B=\frac{5}{2}\±\frac{15}{2}](https://tex.z-dn.net/?f=B%3D%5Cfrac%7B5%7D%7B2%7D%5C%C2%B1%5Cfrac%7B15%7D%7B2%7D)
The solutions are:
![B_1=\frac{5}{2}+\frac{15}{2}=10](https://tex.z-dn.net/?f=B_1%3D%5Cfrac%7B5%7D%7B2%7D%2B%5Cfrac%7B15%7D%7B2%7D%3D10)
![B_2=\frac{5}{2}-\frac{15}{2}=-5](https://tex.z-dn.net/?f=B_2%3D%5Cfrac%7B5%7D%7B2%7D-%5Cfrac%7B15%7D%7B2%7D%3D-5)
For this problem we take the positive solution.
![B=10\ cm](https://tex.z-dn.net/?f=B%3D10%5C%20cm)
Now we substitute the value of B in the equation to find the height h
![h = 10-5](https://tex.z-dn.net/?f=h%20%3D%2010-5)
![h = 5\ cm](https://tex.z-dn.net/?f=h%20%3D%205%5C%20cm)