10.4. a) The heights of a triangle are 3, 4, and 5. Is this triangle a right triangle, an acute triangle, or an obtuse triangle?

b) The same question, if the medians of the triangle are 3, 4, and 5.

## VERY Acute angle
10.4. a) Let \(a, b, c\) be the sides of the triangle, \(h_{a}=3, h_{b}=4, h_{c}=5\) be the heights dropped to these sides, and \(s\) be the area of the triangle. Then \(a h_{a} = b h_{b} = c h_{c} = 2 s\), from which \(a = \frac{2 s}{h_{a}}, b = \frac{2 s}{h_{b}}, c = \frac{2 s}{h_{c}}\), i.e., the given triangle is similar to a triangle with sides \(\frac{1}{h_{a}}, \frac{1}{h_{b}}, \frac{1}{h_{c}}\), which are \(\frac{1}{3}, \frac{1}{4}, \frac{1}{5}\). Since

![](https://cdn.mathpix.com/cropped/2024_05_21_20eb9f6ef2c568605a63g-67.jpg?height=265&width=483&top_left_y=777&top_left_x=129)

a.)

![](https://cdn.mathpix.com/cropped/2024_05_21_20eb9f6ef2c568605a63g-67.jpg?height=313&width=456&top_left_y=758&top_left_x=633)
6)

Fig. 10.4.

\(\left(\frac{1}{3}\right)^{2} > \left(\frac{1}{4}\right)^{2} + \left(\frac{1}{5}\right)^{2}\), the given triangle is obtuse.

b) Let \(m_{a}=3, m_{b}=4, m_{c}=5\) be the medians of the triangle drawn to the sides \(a, b, c\). By extending the triangle to a parallelogram (figure a) and using the fact that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides, we obtain the expressions for the medians of the triangle in terms of its sides:

\[
\begin{gathered}
4 m_{a}^{2} = 2 b^{2} + 2 c^{2} - a^{2} \\
4 m_{b}^{2} = 2 a^{2} + 2 c^{2} - b^{2} \\
4 m_{c}^{2} = 2 a^{2} + 2 b^{2} - c^{2}
\end{gathered}
\]

Solving this system for \(a^{2}, b^{2}\), and \(c^{2}\), we get \(a^{2} = \frac{4}{9} \left(2 m_{b}^{2} + 2 m_{c}^{2} - m_{a}^{2}\right)\), \(b^{2} = \frac{4}{9} \left(2 m_{a}^{2} + 2 m_{c}^{2} - m_{b}^{2}\right)\), \(c^{2} = \frac{4}{9} \left(2 m_{a}^{2} + 2 m_{b}^{2} - m_{c}^{2}\right)\). Substituting \(m_{a}=3\), \(m_{b}=4\), \(m_{c}=5\), we find \(a^{2} = 292 / 9\), \(b^{2} = 208 / 9\), \(c^{2} = 100 / 9\). Since \(a^{2} > \frac{b^{2} + c^{2}}{2}\), the vertex \(A\) lies outside the circle constructed on side \(BC\) as a diameter (figure b), so angle \(A\) is acute. Similarly, we can prove that angles \(B\) and \(C\) are also acute.