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.
how can the dimensions be changed to make the figure possible? select all that apply.
a. decrease the measure of the angle 84°.
b. increase the length of the side opposite the larger angle so it is longer than the side opposite the smaller angle
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.
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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. Here, we have two angles: \(84^{\circ}\) and \(87^{\circ}\), and two sides: \(11\) and \(13\). The angle \(87^{\circ}\) (larger angle) should be opposite the longer side (\(13\)), and \(84^{\circ}\) (smaller angle) opposite the shorter side (\(11\)). But let's analyze each option:

Step2: Analyze Option A

If we decrease the measure of the \(84^{\circ}\) angle, the difference between the angles might adjust, but this doesn't directly fix the side - angle mismatch. Wait, no, let's re - think. The key is the side - angle correspondence. The side opposite \(87^{\circ}\) is \(11\) (wrong, should be longer than the side opposite \(84^{\circ}\)) and the side opposite \(84^{\circ}\) is \(13\) (wrong, should be shorter).

Step3: Analyze Option B

Increase the length of the side opposite the larger angle (\(87^{\circ}\)) so it is longer than the side opposite the smaller angle (\(84^{\circ}\)). Currently, the side opposite \(87^{\circ}\) is \(11\) and opposite \(84^{\circ}\) is \(13\). If we make the side opposite \(87^{\circ}\) longer than the side opposite \(84^{\circ}\), the triangle becomes valid. So this option is correct.

Step4: Analyze Option C

Decrease the length of the side labeled \(11\). If we decrease \(11\) (the side opposite \(87^{\circ}\)), it will be even shorter than the side opposite \(84^{\circ}\) (\(13\)), which is worse. So this is incorrect.

Step5: Analyze Option D

Increase the measure of the \(87^{\circ}\) angle. Increasing the larger angle will make the side - angle mismatch worse (since the side opposite it is still shorter). So this is incorrect.

Step6: Analyze Option E

Increase the length of the side labeled \(13\). The side labeled \(13\) is opposite \(84^{\circ}\) (smaller angle). If we increase \(13\), it will be even longer than the side opposite \(87^{\circ}\) (\(11\)), worsening the mismatch. So this is incorrect. Wait, no, wait. Wait the side labels: looking at the figure, the side of length \(13\) is adjacent to \(87^{\circ}\), and the side of length \(11\) is adjacent to \(84^{\circ}\). Wait, maybe I mis - identified the opposite sides. Let's re - visualize the triangle: the two angles \(84^{\circ}\) and \(87^{\circ}\) are at the same vertex, and the sides \(11\) and \(13\) are the other two sides. The side opposite \(84^{\circ}\) is \(13\), and opposite \(87^{\circ}\) is \(11\). So to make the triangle valid, the side opposite the larger angle (\(87^{\circ}\)) should be longer than the side opposite the smaller angle (\(84^{\circ}\)). So option B: increase the length of the side opposite the larger angle (so that it's longer than the side opposite the smaller angle) is correct. Also, if we decrease the measure of the \(87^{\circ}\) angle? Wait no, the first part of the question: "why the figure cannot have the labeled dimensions" is because in a triangle, the larger angle must be opposite the longer side. Here, \(87^{\circ}>84^{\circ}\), but the side opposite \(87^{\circ}\) (let's say side \(a\)) is \(11\), and the side opposite \(84^{\circ}\) (side \(b\)) is \(13\), so \(a < b\) while the angle opposite \(a\) is larger. So to fix it:

- Option B: Make the side opposite the larger angle (\(87^{\circ}\)) longer than the side opposite the smaller angle (\(84^{\circ}\)): correct.

- Option A: Decrease the measure of the \(84^{\circ}\) angle: if we decrease \(84^{\circ}\), then the difference between \(87^{\circ}\) and the new angle of \(84^{\circ}\) (say \(84^{\circ}-\Delta\)) increases, but also, the side opposite the new smaller angle (now less than \(84^{\circ}\)) should be shorter. But maybe if we decrease \(84^{\circ}\) enough, the side opposite \(87^{\circ}\) (11) can be longer than the side opposite the new smaller angle. Wait, maybe I made a mistake earlier. Let's use the triangle angle - side relationship: in a triangle, \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) (Law of Sines). Let \(A = 87^{\circ}\), \(a = 11\); \(B = 84^{\circ}\), \(b = 13\). Then \(\frac{11}{\sin87^{\circ}}\approx\frac{11}{0.9986}\approx11.02\), \(\frac{13}{\sin84^{\circ}}\approx\frac{13}{0.9945}\approx13.07\). Since \(\frac{a}{\sin A}<\frac{b}{\sin B}\), the triangle is invalid.

If we decrease the measure of angle \(A = 87^{\circ}\) (wait, option A is decrease \(84^{\circ}\)). Wait, no, option A: Decrease the measure of the angle \(84^{\circ}\). Let's say we decrease \(84^{\circ}\) to \(80^{\circ}\). Then \(A = 87^{\circ}\), \(B = 80^{\circ}\). Then \(\frac{a}{\sin A}=\frac{11}{\sin87^{\circ}}\approx11.02\), \(\frac{b}{\sin B}=\frac{13}{\sin80^{\circ}}\approx\frac{13}{0.9848}\approx13.2\). Still \(\frac{a}{\sin A}<\frac{b}{\sin B}\). But if we increase the length of the side opposite the larger angle (option B): let's say we increase \(a\) (side opposite \(87^{\circ}\)) to \(14\). Then \(\frac{14}{\sin87^{\circ}}\approx\frac{14}{0.9986}\approx14.02\), \(\frac{13}{\sin84^{\circ}}\approx13.07\). Now \(\frac{a}{\sin A}>\frac{b}{\sin B}\), so the triangle is valid.

Wait, maybe I mis - analyzed option A. Let's re - check the options:

Option A: Decrease the measure of the angle \(84^{\circ}\). If we decrease \(84^{\circ}\), then the third angle \(C=180 - 87 - (84 - x)=9 + x\) (where \(x>0\)). Now, the side opposite \(87^{\circ}\) is \(11\), side opposite \((84 - x)^{\circ}\) is \(13\), side opposite \((9 + x)^{\circ}\) is \(c\). By Law of Sines, \(\frac{11}{\sin87^{\circ}}=\frac{13}{\sin(84 - x)^{\circ}}=\frac{c}{\sin(9 + x)^{\circ}}\). If we decrease \(x\) (decrease \(84^{\circ}\) to a smaller angle), \(\sin(84 - x)^{\circ}\) decreases (since \(84 - x<84\) and \(\sin\) is increasing on \([0,90]\)). So \(\frac{13}{\sin(84 - x)^{\circ}}\) increases. \(\frac{11}{\sin87^{\circ}}\approx11.02\). If we decrease \(84^{\circ}\) until \(\frac{13}{\sin(84 - x)^{\circ}} = 11.02\), then \(\sin(84 - x)^{\circ}=\frac{13}{11.02}\approx1.179\), which is impossible (since \(\sin\theta\leq1\)). So option A is wrong.

Option B: Increase the length of the side opposite the larger angle ( \(87^{\circ}\)) so it is longer than the side opposite the smaller angle ( \(84^{\circ}\)). Let's say we increase the side opposite \(87^{\circ}\) (originally \(11\)) to \(14\). Then \(\frac{14}{\sin87^{\circ}}\approx14.02\), \(\frac{13}{\sin84^{\circ}}\approx13.07\). Now \(\frac{14}{\sin87^{\circ}}>\frac{13}{\sin84^{\circ}}\), so by Law of Sines, the triangle is valid. So option B is correct.

Option C: Decrease the length of the side labeled \(11\): this will make the side opposite \(87^{\circ}\) even shorter, worsening the problem.

Option D: Increase the measure of the angle \(87^{\circ}\): this will make the angle opposite the shorter side even larger, worsening the problem.

Option E: Increase the length of the side labeled \(13\): this side is opposite \(84^{\circ}\) (smaller angle), making it longer will worsen the side - angle mismatch.

Wait, maybe I made a mistake in side - angle correspondence. Let's look at the figure again (from the diagram: the two angles \(84^{\circ}\) and \(87^{\circ}\) are at the same vertex, and the sides \(11\) and \(13\) are the other two sides. So the side adjacent to both angles is the common side, and the other two sides: one is \(11\) (opposite \(87^{\circ}\)) and one is \(13\) (opposite \(84^{\circ}\)). So the correct way to fix it is to make the side opposite the larger angle (\(87^{\circ}\)) longer than the side opposite the smaller angle (\(84^{\circ}\)), which is option B. Also, if we decrease the measure of the \(87^{\circ}\) angle (but there is no option for that). Wait, the original analysis was wrong. Let's start over.

The triangle inequality and angle - side relationship: in any triangle, the larger angle is opposite the longer side. So if we have two angles, \( \angle1 = 87^{\circ}\) and \( \angle2=84^{\circ}\), then the side opposite \( \angle1\) (let's call it \(s_1\)) should be longer than the side opposite \( \angle2\) ( \(s_2\)). Currently, \(s_1 = 11\) and \(s_2 = 13\), so \(s_1<s_2\) which violates the angle - side relationship.

To fix this:

- We can increase \(s_1\) (side opposite \(87^{\circ}\)) so that \(s_1>s_2\) (option B: "Increase the length of the side opposite the larger angle so it is longer than the side opposite the smaller angle").

- Or we can decrease \(s_2\) (but there is no option for that).

- Or we can decrease the measure of \( \angle1\) ( \(87^{\circ}\)) so that \( \angle1<\angle2\) and then \(s_1<s_2\) would be correct, but there is no option for that.

- Or we can increase the measure of \( \angle2\) ( \(84^{\circ}\)) so that \( \angle2>\angle1\) and then \(s_2>s_1\) would be correct, but there is no option for that.

Wait, the options:

A. Decrease the measure of the angle \(84^{\circ}\): as we saw, this makes \( \angle2\) smaller, so \( \angle1-\angle2\) larger, and \(s_1 = 11\), \(s_2 = 13\) still has \(s_1<s_2\) with \( \angle1>\angle2\), so worse.

B. Increase the length of the side opposite the larger angle so it is longer than the side opposite the smaller angle: this fixes \(s_1>s_2\) for \( \angle1>\angle2\), correct.

C. Decrease the length of the side labeled \(11\): makes \(s_1\) smaller, worse.

D. Increase the measure of the angle \(87^{\circ}\): makes \( \angle1\) larger, so \(s_1\) should be even longer, but \(s_1 = 11\) is still small, worse.

E. Increase the length of the side labeled \(13\): makes \(s_2\) larger, so \(s_1 = 11\) is even smaller relative to \(s_2\), worse.

So the correct option is B. Wait, but maybe I missed something. Wait the first part: "Explain why the figure cannot have the labeled dimensions" is because the larger angle (\(87^{\circ}\)) is opposite the shorter side (\(11\)) and the smaller angle (\(84^{\circ}\)) is opposite the longer side (\(13\)), violating the angle - side relationship in a triangle. Then "how can the dimensions be changed to make the figure possible": we need to make the side opposite the larger angle longer than the side opposite the smaller angle, which is option B. Also, is there another correct option? Let's re - check option A: "Decrease the measure of the angle \(84^{\circ}\)". If we decrease \(84^{\circ}\) to, say, \(80^{\circ}\), then the third angle is \(180 - 87 - 80=13^{\circ}\). Now, using Law of Sines: \(\frac{11}{\sin87^{\circ}}\approx11.02\), \(\frac{13}{\sin80^{\circ}}\approx13.2\), \(\frac{c}{\sin13^{\circ}}\approx\frac{c}{0.225}\). For the triangle to be valid, \(\frac{11}{\sin87^{\circ}}+\frac{13}{\sin80^{\circ}}>c\), \(\frac{11}{\sin87^{\circ}}+c>\frac{13}{\sin80^{\circ}}\), \(\frac{13}{\sin80^{\circ}}+c>\frac{11}{\sin87^{\circ}}\). But the angle - side relationship: the largest angle is \(87^{\circ}\), so its opposite side should be the longest. But \(11\) (opposite \(87^{\circ}\)) is shorter than \(13\) (opposite \(80^{\circ}\)), so it's still invalid. So option A is wrong. So only option B is correct? Wait, no, maybe the original figure has the two angles at the same vertex, and the sides \(11\) and \(13\) are the two sides forming the angles. Wait, maybe it's a quadrilateral? No, the diagram shows a triangle divided by a diagonal. Wait, the figure is a quadrilateral with a diagonal, making two triangles. In a quadrilateral, the sum of angles and sides: but the problem is about the triangle formed.

Wait, the key is the triangle angle - side rule: larger angle opposite longer side. So to make the triangle possible, we need the side opposite the larger angle to be longer than the side opposite the smaller angle. So option B is correct. Also, if we increase the length of the side opposite the larger angle (option B) or decrease the length of the side opposite the smaller angle (but there is no option for that) or adjust the angles. But among the given options, only option B is correct. Wait, maybe I made a mistake earlier. Let's confirm with the Law of Sines. In a triangle, \(\frac{a}{\sin A}=\frac{b}{\sin B}\). If \(A > B\), then \(a > b\). Currently, \(A = 87^{\circ}\), \(a = 11\); \(B = 84^{\circ}\), \(b = 13\). So \(\frac{11}{\sin87^{\circ}}<\frac{13}{\sin84^{\circ}}\) (since \(11\times\sin84^{\circ}<13\times\sin87^{\circ}\): \(11\times0.9945\approx10.94\), \(13\times0.9986\approx12.98\), \(10.94 < 12.98\)). So to make \(\frac{a}{\sin A}=\frac{b}{\sin B}\), we can either increase \(a\) (so that \(a\times\sin B>b\times\sin A\)) or decrease \(b\) or adjust the angles. Since increasing \(a\) (side opposite \(A = 87^{\circ}\)) so that \(a>b\) (option B) is a valid way, option B is correct.

Answer:

B. Increase the length of the side opposite the larger angle so it is longer than the side opposite the smaller angle