Question

(4-27) ben thinks that the slope ratio for this triangle is 7/10. carlissa thinks the ratio is 10/7. who is correct? explain your thinking. (4-28) a. thalia did not have a tool to help her find the slope angle in the triangle at right. however, she claims that the slope angle has to be more than 45°. do you agree with thalia?

Explanation:

Step1: Recall slope - ratio formula

The slope ratio (slope) of a right - triangle is given by $m=\frac{\text{rise}}{\text{run}}$, where the rise is the vertical side and the run is the horizontal side.

Step2: Identify rise and run for the first triangle

For the first triangle with vertical side $7$ and horizontal side $10$, the slope ratio $m = \frac{7}{10}$. So Ben is correct.

Step3: Recall relationship between slope and angle for the second triangle

The slope of a right - triangle is related to the angle $\theta$ (the non - right angle) by $\tan\theta=\frac{\text{rise}}{\text{run}}$. For the second triangle, $\text{rise}=6$ and $\text{run}=4$, so $\tan\theta=\frac{6}{4} = 1.5$.

Step4: Compare with $\tan45^{\circ}$

We know that $\tan45^{\circ}=1$. Since $\tan\theta = 1.5>1=\tan45^{\circ}$ and the tangent function $y = \tan x$ is increasing for $0^{\circ}45^{\circ}$. So we agree with Thalia.

Answer:

4 - 27: Ben is correct because the slope ratio of a right - triangle is $\frac{\text{vertical side}}{\text{horizontal side}}=\frac{7}{10}$.
4 - 28: Agree. Since $\tan\theta=\frac{\text{rise}}{\text{run}}=\frac{6}{4}=1.5$ and $\tan45^{\circ}=1$, and the tangent function is increasing for $0^{\circ}<\theta<90^{\circ}$, the slope angle $\theta>45^{\circ}$.