Question

a student draws the figure to the right. explain why the figure cannot have the labeled dimensions. then describe a way to change the dimensions to make the figure possible. figure not drawn to scale. c. decrease the length of the side labeled 11. d. increase the measure of the angle 87°. e. increase the length of the side labeled 13. f. decrease the length of the side opposite the smaller angle so it is shorter than the side opposite the larger angle. g. switch the labels of the two sides labeled 13 and 11. help me solve this view an example get more help

Explanation:

Step1: Recall Triangle Side - Angle Relationship

In a triangle, the larger angle is opposite the longer side, and the smaller angle is opposite the shorter side (Law of Sines: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ implies that if $A > B$, then $a > b$).

Step2: Analyze Given Angles and Sides

In the triangle formed by the diagonal, we have one angle of $87^{\circ}$ and another angle of $84^{\circ}$. Since $87^{\circ}>84^{\circ}$, the side opposite $87^{\circ}$ should be longer than the side opposite $84^{\circ}$. The side opposite $87^{\circ}$ is 11, and the side opposite $84^{\circ}$ is 13. But $11 < 13$, which violates the side - angle relationship.

Step3: Evaluate Each Option

• Option C: Decreasing the length of the side labeled 11 would make it even shorter than 13, which is still a violation.
• Option D: Increasing the measure of the angle $87^{\circ}$ would make the angle opposite 11 larger, and according to the side - angle relationship, the side opposite (11) should be longer than the side opposite $84^{\circ}$ (13). But 11 is less than 13, so increasing the $87^{\circ}$ angle would require 11 to be longer, but we can't change the side length this way directly. Wait, no, actually, if we consider the relationship, to make the figure possible, we can switch the labels of the two sides labeled 13 and 11 (Option G), or increase the length of the side opposite the larger angle (11) or decrease the length of the side opposite the smaller angle (13), or adjust the angles. But among the options, let's re - evaluate:

Wait, the correct approach is: The side opposite the larger angle ($87^{\circ}$) should be longer than the side opposite the smaller angle ($84^{\circ}$). Currently, side opposite $87^{\circ}$ is 11, side opposite $84^{\circ}$ is 13. So 11 (opposite $87^{\circ}$) < 13 (opposite $84^{\circ}$) which is wrong. So to fix this, we can switch the labels of 11 and 13 (Option G), or increase the length of 11 (not an option here? Wait, the options:
Option G: Switch the labels of the two sides labeled 13 and 11. If we switch them, then the side opposite $87^{\circ}$ is 13 and the side opposite $84^{\circ}$ is 11, and since $87^{\circ}>84^{\circ}$, $13 > 11$, which satisfies the side - angle relationship.
Option C: Decreasing 11 makes it worse. Option D: Increasing the $87^{\circ}$ angle: if we increase the $87^{\circ}$ angle, the side opposite (11) should be longer, but we can't change the side length by changing the angle. Option E: Increasing the length of 13: if we increase 13, the side opposite $84^{\circ}$ (13) would be longer, but the side opposite $87^{\circ}$ (11) is still shorter. So the correct option is G. Also, let's check again:
The problem is that the side opposite the larger angle is shorter than the side opposite the smaller angle. So to correct it, we can switch the side labels (11 and 13) so that the longer side is opposite the larger angle.

Answer:

G. Switch the labels of the two sides labeled 13 and 11.