the angles of a triangle are in the ratio 3:5:4 calculate the size of each angle
+26393 the angles of a triangle are in the ratio 3:5:4 calculate the size of each angle
\(\begin{array}{rcll} \text{Let } \alpha &=& \text{angle}_1 \\ \text{Let } \beta &=& \text{angle}_2 \\ \text{Let } \gamma &=& \text{angle}_3 \\ \gamma &=& 180^{\circ}-(\alpha+\beta) \\ \end{array} \)
\(\begin{array}{|rcll|} \hline && \alpha : \beta : \gamma \\ &=& \alpha : \beta : 180^{\circ}-(\alpha+\beta) \\ &=& 3:5:4 \\ \hline \end{array} \)
\(\begin{array}{|lrcll|} \hline (1): & \frac{\alpha}{\beta} &=& \frac{3}{5} \\ & \alpha &=& \frac{3}{5}\cdot \beta \\\\ (2): & \frac{\alpha}{\gamma} = \frac{\alpha} { 180^{\circ}-(\alpha+\beta) } &=& \frac{3}{4} \\ & \frac{\alpha} { 180^{\circ}-(\alpha+\beta) } &=& \frac{3}{4} \\ & \frac{4}{3} \cdot \alpha &=& 180^{\circ}-(\alpha+\beta) \quad & | \quad \alpha = \frac{3}{5}\cdot \beta \\ & \frac{4}{3} \cdot \frac{3}{5}\cdot \beta &=& 180^{\circ}-(\frac{3}{5}\cdot \beta +\beta) \\ & \frac{4}{5} \cdot \beta &=& 180^{\circ}-\frac{8}{5}\cdot \beta \\ & \frac{4}{5} \cdot \beta +\frac{8}{5}\cdot \beta &=& 180^{\circ} \\ & \frac{12}{5} \cdot \beta &=& 180^{\circ} \\ & \beta &=& 180^{\circ}\cdot \frac{5}{12} \\ & \mathbf{ \beta } & \mathbf{=} & \mathbf{ 75^{\circ} } \\ \hline \end{array} \)
\(\begin{array}{|lrcll|} \hline & \alpha &=& \frac{3}{5}\cdot \beta \quad & | \quad \mathbf{ \beta = 75^{\circ} } \\ & \alpha &=& \frac{3}{5}\cdot 75^{\circ} \\ & \mathbf{ \alpha } & \mathbf{=} & \mathbf{ 45^{\circ} } \\ \\ & \gamma &=& 180^{\circ}-(\alpha+\beta) \quad & | \quad \mathbf{ \alpha = 45^{\circ} } \qquad \mathbf{ \beta = 75^{\circ} } \\ & \gamma &=& 180^{\circ}-( 45^{\circ}+75^{\circ} ) \\ & \gamma &=& 180^{\circ}-120^{\circ} \\ & \mathbf{ \gamma } & \mathbf{=} & \mathbf{ 60^{\circ} } \\ \hline \end{array} \)
The angles of the triangle are \(45^{\circ},\ 75^{\circ},\ 60^{\circ}\)
+129899
3 : 4 : 5 means that there are 3 + 4 + 5 = 12 equal parts
And the angles of a triangle sum to 180
So....one of these angles is 3/12 of this = 3/12 * 180 = 1/4 * 180 = 45°
And another of the angles is 4/12 of this = 4/12 * 180 = 1/3 * 180 = 60°
And the last angle must be 180 - 45 - 60 = 75°
+229 I would just take the 12 parts, and divide 180 the correct amount of times. That 1st answer is over complicated. LOL!
