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# Ratio

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the angles of a triangle are in the ratio 3:5:4 calculate the size of each angle

Guest Apr 20, 2017
#1
+20033
+1

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 $$45^{\circ},\ 75^{\circ},\ 60^{\circ}$$

heureka  Apr 20, 2017
#2
+90056
+2

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°

CPhill  Apr 20, 2017
#3
+229
+1

I would just take the 12 parts, and divide 180 the correct amount of times. That 1st answer is over complicated. LOL!

liveevillevi  Apr 20, 2017