Question

(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?
b. lyra was trying to find the slope ratio for the triangle at right, and she says the answer is $\frac{delta y}{delta x}=2.675$. isiah claims that cannot be correct. who is right? how do you know?
c. without finding the actual value, what information do you know about x in the diagram at right?

Explanation:

Step1: Recall slope - angle relationship

The slope ratio is $\tan\theta$, where $\theta$ is the slope - angle. For a right - triangle, if the opposite side is $y$ and the adjacent side is $x$, the slope ratio is $\frac{\Delta y}{\Delta x}=\tan\theta$.

Step2: Analyze part a

In a right - triangle with opposite side $y = 6$ and adjacent side $x = 4$, the slope ratio $m=\frac{y}{x}=\frac{6}{4}=1.5$. Since $\tan45^{\circ}=1$ and the tangent function $y = \tan\theta$ is increasing for $0^{\circ}<\theta<90^{\circ}$, when $\tan\theta=1.5>1$, $\theta > 45^{\circ}$. So, Thalia is correct.

Step3: Analyze part b

For a right - triangle with an angle $\theta = 19^{\circ}$, the slope ratio $\frac{\Delta y}{\Delta x}=\tan\theta$. We know that $\tan19^{\circ}\approx0.3443$. Since $0.3443
eq2.675$, Isiah is right.

Step4: Analyze part c

In a right - triangle with an angle $\theta = 76^{\circ}$, the slope ratio $\frac{\text{opposite}}{\text{adjacent}}=\tan76^{\circ}\approx4.0108$. Since $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, and $\theta = 76^{\circ}>45^{\circ}$, we know that $\tan\theta>1$. So the length of the side opposite the $76^{\circ}$ angle is greater than the length of the side adjacent to the $76^{\circ}$ angle. That is, the side opposite the $76^{\circ}$ angle is greater than $x$.

Answer:

a. Yes
b. Isiah is right because $\tan19^{\circ}\approx0.3443
eq2.675$.
c. The side opposite the $76^{\circ}$ angle is greater than $x$.