Question

determine the measure of side d in the following diagram. round your answer to the nearest tenth. select one: a. 7.1 b. 8.1 c. 9.1 d. 6.1

Explanation:

Step1: Find the common side (let's call it \( x \))

In the right - triangle with legs 3.8 and \( x \), and angle \( 28^{\circ} \), we know that \( \sin(28^{\circ})=\frac{3.8}{x} \) (opposite side over hypotenuse). Wait, no, actually, in the right - triangle, if the right angle is at the bottom - left, and the angle at the bottom - right is \( 28^{\circ} \), then \( \sin(28^{\circ})=\frac{3.8}{x} \) is incorrect. Let's correct: In the right - triangle, the side of length 3.8 is opposite the \( 28^{\circ} \) angle, and \( x \) is the hypotenuse of the right - triangle. So by the definition of sine: \( \sin(28^{\circ})=\frac{3.8}{x} \), then \( x = \frac{3.8}{\sin(28^{\circ})} \).
\( \sin(28^{\circ})\approx0.4695 \), so \( x=\frac{3.8}{0.4695}\approx8.1 \) (this is the length of the common side between the two triangles).

Step2: Use the Law of Sines in the upper triangle

In the upper triangle, we have an angle of \( 55^{\circ} \), and the angle opposite to side \( d \) can be found as follows: The angle adjacent to \( 67^{\circ} \) and \( 28^{\circ} \) is a straight angle, so the angle in the upper triangle at the bottom - right is \( 180^{\circ}-67^{\circ}=113^{\circ} \)? Wait, no. Wait, the common side is \( x \approx8.1 \). Wait, actually, the upper triangle has angles: one angle is \( 55^{\circ} \), another angle: since the angle at the bottom - right of the upper triangle is \( 180^{\circ}-67^{\circ}=113^{\circ} \)? No, wait, the angle between the common side and the side \( d \) in the upper triangle: let's re - examine. The lower triangle is a right - triangle with right angle, angle \( 28^{\circ} \), so the third angle is \( 90^{\circ}-28^{\circ}=62^{\circ} \). Wait, maybe a better approach: The common side (let's call it \( c \)) can be found from the right - triangle: in the right - triangle, \( \cos(28^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}} \), no, wait the right - triangle has legs: one leg is 3.8, the other leg is, say, \( y \), and hypotenuse \( c \). Wait, the angle at the bottom - right of the right - triangle is \( 28^{\circ} \), so \( \sin(28^{\circ})=\frac{3.8}{c} \), so \( c=\frac{3.8}{\sin(28^{\circ})}\approx\frac{3.8}{0.4695}\approx8.1 \). Now, in the upper triangle, we have a triangle with angle \( 55^{\circ} \), angle at the bottom - right: since the angle adjacent to \( 67^{\circ} \) and \( 28^{\circ} \) is a straight line, the angle in the upper triangle at the bottom - right is \( 180^{\circ}-(67^{\circ} + 28^{\circ})=85^{\circ} \)? No, wait, the angle between the common side \( c \) and the side \( d \) in the upper triangle: the angle at the vertex where the two triangles meet (the bottom - right vertex of the upper triangle) is \( 180^{\circ}-67^{\circ}=113^{\circ} \)? I think I made a mistake earlier. Let's use the Law of Sines correctly.

Wait, the upper triangle: we know one side (the common side \( c\approx8.1 \)) and the angle opposite to it. Wait, the angle opposite to \( c \) in the upper triangle: the sum of angles in a triangle is \( 180^{\circ} \). The upper triangle has angles: \( 55^{\circ} \), and the angle at the bottom - right: let's see, the angle between the common side and the side \( d \) is \( 180^{\circ}-67^{\circ}=113^{\circ} \)? No, the angle at the bottom - right of the upper triangle: the lower triangle has an angle of \( 28^{\circ} \) at the bottom - right, and the upper triangle's angle at the bottom - right is supplementary to \( 67^{\circ} \)? Wait, the diagram shows that the angle between the two sides at the bottom - right is \( 67^{\circ} \) (the curved angle) and \( 28^{\circ} \) (the other curved angle), so the angle in the upper triangle at the bottom - right is \( 180^{\circ}-67^{\circ}=113^{\circ} \)? No, that can't be. Wait, maybe the upper triangle has angles: \( 55^{\circ} \), and the angle opposite to \( d \) is \( 180^{\circ}-55^{\circ}-\theta \), where \( \theta \) is the angle at the bottom - right. Wait, let's start over.

In the right - triangle (lower triangle):
- Right angle (\( 90^{\circ} \)), angle \( 28^{\circ} \), so the third angle is \( 90 - 28=62^{\circ} \).
- The side of length 3.8 is opposite the \( 28^{\circ} \) angle. Let the hypotenuse of the right - triangle (the common side between the two triangles) be \( c \). Then by sine: \( \sin(28^{\circ})=\frac{3.8}{c} \), so \( c=\frac{3.8}{\sin(28^{\circ})}\approx\frac{3.8}{0.4695}\approx8.1 \).

Now, in the upper triangle:
- We have a triangle with one side \( c = 8.1 \), the angle opposite to \( c \): let's find the angles. The angle at the top is \( 55^{\circ} \), the angle at the bottom - right: since the angle adjacent to \( 67^{\circ} \) and \( 28^{\circ} \) is a straight line, the angle in the upper triangle at the bottom - right is \( 180-(67 + 28)=85^{\circ} \)? No, wait, the angle at the bottom - right of the upper triangle: the lower triangle has an angle of \( 28^{\circ} \) at the bottom - right, and the upper triangle's angle at the bottom - right is \( 67^{\circ} \)? No, the diagram shows two angles at the bottom - right vertex: \( 67^{\circ} \) (the larger angle) and \( 28^{\circ} \) (the smaller angle), so the angle in the upper triangle at the bottom - right is \( 67^{\circ} \)? Wait, no, the two triangles share a common side (the horizontal side). The lower triangle is a right - triangle with right angle, side 3.8, angle \( 28^{\circ} \). The upper triangle has angle \( 55^{\circ} \) at the top, and angle \( 180 - 55 - \alpha \), where \( \alpha \) is the angle at the bottom - right. Wait, maybe the Law of Sines is \( \frac{d}{\sin(\alpha)}=\frac{c}{\sin(55^{\circ})} \), where \( \alpha \) is the angle opposite to \( c \) in the upper triangle.

Wait, actually, the correct approach is:

1. Find the length of the common side (let's call it \( c \)) from the right - triangle:
In the right - triangle, \( \sin(28^{\circ})=\frac{3.8}{c} \) (since 3.8 is opposite the \( 28^{\circ} \) angle, and \( c \) is the hypotenuse).
So \( c=\frac{3.8}{\sin(28^{\circ})}\approx\frac{3.8}{0.4695}\approx8.1 \).

2. Now, in the upper triangle, we can use the Law of Sines. Wait, the upper triangle has angles: let's find the angles. The angle at the top is \( 55^{\circ} \), the angle at the bottom - right: since the sum of angles in a triangle is \( 180^{\circ} \), and we know that the angle opposite to \( c \) (which is \( 8.1 \)): wait, no, maybe the upper triangle has angles \( 55^{\circ} \), and the angle opposite to \( d \) is \( 180 - 55 - \beta \), where \( \beta \) is the angle at the bottom - right. Wait, I think I messed up the angle identification. Let's look at the answer options. The common side we found is approximately 8.1, and in the upper triangle, if we use the Law of Sines, and assume that the angle opposite to \( d \) is equal to the angle opposite to 3.8 in the right - triangle (which is \( 28^{\circ} \))? No, that doesn't make sense. Wait, maybe the two triangles are related such that the upper triangle has a side equal to the hypotenuse of the right - triangle, and we can use the Law of Sines.

Wait, the correct way:

- In the right - triangle: \( \sin(28^{\circ})=\frac{3.8}{c}\implies c = \frac{3.8}{\sin(28^{\circ})}\approx8.1 \) (this is the length of the common side).

- In the upper triangle, we have a triangle with side \( c = 8.1 \), angle \( 55^{\circ} \), and we want to find \( d \). Wait, maybe the upper triangle has angles: \( 55^{\circ} \), and the angle opposite to \( d \) is \( 180 - 55 - 67=58^{\circ} \)? No, this is getting confusing. Wait, the answer option b is 8.1, but that's the length of the common side. Wait, no, maybe I made a mistake in the angle. Let's re - examine the diagram. The lower triangle is a right - triangle with right angle, side 3.8, angle \( 28^{\circ} \). The upper triangle has angle \( 55^{\circ} \) at the top, and the angle at the bottom - right is \( 67^{\circ} \). Wait, the sum of angles in the upper triangle: \( 55 + 67+\gamma=180\implies\gamma = 180-(55 + 67)=58^{\circ} \). Now, using the Law of Sines: \( \frac{d}{\sin(67^{\circ})}=\frac{c}{\sin(55^{\circ})} \), where \( c \) is the side opposite to \( 55^{\circ} \). Wait, \( c \) is the length we found from the right - triangle: \( c=\frac{3.8}{\sin(28^{\circ})}\approx8.1 \). Then \( d=\frac{c\times\sin(67^{\circ})}{\sin(55^{\circ})} \). \( \sin(67^{\circ})\approx0.9205 \), \( \sin(55^{\circ})\approx0.8192 \). So \( d=\frac{8.1\times0.9205}{0.8192}\approx\frac{7.456}{0.8192}\approx9.1 \). Wait, now I'm confused. Wait, maybe the common side is not the hypotenuse. Let's try cosine in the right - triangle. In the right - triangle, \( \cos(28^{\circ})=\frac{\text{adjacent}}{c} \), no, the side of length 3.8 is opposite \( 28^{\circ} \), so \( \sin(28^{\circ})=\frac{3.8}{c}\implies c=\frac{3.8}{\sin(28^{\circ})}\approx8.1 \). Then in the upper triangle, using Law of Sines: \( \frac{d}{\sin(180 - 55 - (180 - 67))}=\frac{c}{\sin(55^{\circ})} \). Wait, the angle at the bottom - right of the upper triangle is \( 67^{\circ} \) (since the angle marked \( 67^{\circ} \) is in the upper triangle's corner). Wait, the two triangles share a common side (the horizontal side). The lower triangle: right angle, side 3.8, angle \( 28^{\circ} \), so the horizontal side (let's call it \( b \)) is \( b = 3.8\times\cot(28^{\circ}) \). \( \cot(28^{\circ})=\frac{\cos(28^{\circ})}{\sin(28^{\circ})}\approx\frac{0.8829}{0.4695}\approx1.88 \), so \( b = 3.8\times1.88\approx7.1 \). No, this is not working. Wait, let's look at the answer options. The correct answer is likely c. 9.1. Wait, let's do it properly.

1. Find the length of the common side (let's call it \( x \)) from the right - triangle:
In the right - triangle, \( \sin(28^{\circ})=\frac{3.8}{x}\implies x=\frac{3.8}{\sin(28^{\circ})}\approx\frac{3.8}{0.4695}\approx8.1 \).

2. In the upper triangle, we have angles: \( 55^{\circ} \), and the angle at the bottom - right is \( 180^{\circ}-67^{\circ}=113^{\circ} \)? No, the sum of angles in a triangle is \( 180^{\circ} \), so the third angle is \( 180-(55 + 113)=12^{\circ} \)? That can't be. I think I misidentified the angles. Let's assume that the upper triangle has angles \( 55^{\circ} \), and the angle opposite to \( x \) (the common side) is \( 180 - 55 - \alpha \), where \( \alpha \) is the angle at the bottom - right. Wait, maybe the diagram is a triangle with a common side, and the lower triangle is right - angled, with side 3.8, angle \( 28^{\circ} \), so the common side (let's call it \( a \)) is \( a=\frac{3.8}{\sin(28^{\circ})}\approx8.1 \). Then in the upper triangle, using the Law of Sines: \( \frac{d}{\sin(67^{\circ})}=\frac{a}{\sin(55^{\circ})} \). So \( d=\frac{a\times\sin(67^{\circ})}{\sin(55^{\circ})} \). \( a\approx8.1 \), \( \sin(67^{\circ})\approx0.9205 \), \( \sin(55^{\circ})\approx0.8192 \). Then \( d=\frac{8.1\times0.9205}{0.8192}\approx\frac{7.456}{0.8192}\approx9.1 \).

Answer:

c. 9.1