Math / Geometry

QuestionBrian and Lashonda are standing on a riverbank, 180 meters apart, at points $A$ and $B$ respectively. (See the figure below.) They want to know the distance from Español
Brian to a house located across the river at point $C$ . Brian measures angle $A$ (angle $B A C$ ) to be $49^{\circ}$ , and Lashonda measures angle $B$ (angle $A B C$ ) to be $73^{\circ}$ . What is the distance from Brian to the house? Round your answer to the nearest tenth of a meter. $\square$ meters

Studdy Solution

STEP 1

What is this asking? We need to find out how far Brian is from a house across the river, using the angles they measured and the distance between Brian and Lashonda. Watch out! Don't forget to use the Law of Sines correctly, and make sure to round your final answer to the nearest tenth!

STEP 2

1. Understand the triangle
2. Apply the Law of Sines
3. Solve for the distance

STEP 3

Alright, let's picture this: Brian and Lashonda are standing on the riverbank, and the house is across the river.
We've got a triangle here, with points $A$ , $B$ , and $C$ .
Brian is at $A$ , Lashonda is at $B$ , and the house is at $C$ .
The distance between $A$ and $B$ is 180 meters.

STEP 4

We know angle $A$ is $49^\circ$ and angle $B$ is $73^\circ$ .
To find angle $C$ , we use the fact that the sum of angles in a triangle is $180^\circ$ .
So, we calculate:
$C = 180^\circ - 49^\circ - 73^\circ$

STEP 5

Plugging in the numbers, we find:
$C = 58^\circ$

STEP 6

The Law of Sines tells us that the ratio of the length of a side to the sine of its opposite angle is the same for all sides of a triangle.
So, we have:
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ where $a$ , $b$ , and $c$ are the sides opposite angles $A$ , $B$ , and $C$ respectively.

STEP 7

We want to find the distance from Brian to the house, which is side $b$ (opposite angle $B$ ).
We know:
- $a = 180$ meters (distance between Brian and Lashonda) - $\sin A = \sin 49^\circ$ - $\sin B = \sin 73^\circ$

STEP 8

Using the Law of Sines, we set up the equation:
$\frac{180}{\sin 58^\circ} = \frac{b}{\sin 73^\circ}$

STEP 9

Let's solve for $b$ by multiplying both sides by $\sin 73^\circ$ :
$b = \frac{180 \cdot \sin 73^\circ}{\sin 58^\circ}$

STEP 10

Now, let's plug in the values for the sines:
- $\sin 73^\circ \approx 0.9563$ - $\sin 58^\circ \approx 0.8480$

STEP 11

Substitute these into the equation:
$b = \frac{180 \cdot 0.9563}{0.8480}$

STEP 12

Calculate the value:
$b \approx \frac{172.134}{0.8480} \approx 203.0$

STEP 13

The distance from Brian to the house is approximately 203.0 meters.

Was this helpful?

Studdy Solution

STEP 1

What is this asking? We need to find out how far Brian is from a house across the river, using the angles they measured and the distance between Brian and Lashonda. Watch out! Don't forget to use the Law of Sines correctly, and make sure to round your final answer to the nearest tenth!

STEP 2

1. Understand the triangle
2. Apply the Law of Sines
3. Solve for the distance

STEP 3

Alright, let's picture this: Brian and Lashonda are standing on the riverbank, and the house is across the river.
We've got a triangle here, with points $A$ , $B$ , and $C$ .
Brian is at $A$ , Lashonda is at $B$ , and the house is at $C$ .
The distance between $A$ and $B$ is 180 meters.

STEP 4

We know angle $A$ is $49^\circ$ and angle $B$ is $73^\circ$ .
To find angle $C$ , we use the fact that the sum of angles in a triangle is $180^\circ$ .
So, we calculate:
$C = 180^\circ - 49^\circ - 73^\circ$

STEP 5

Plugging in the numbers, we find:
$C = 58^\circ$

STEP 6

The Law of Sines tells us that the ratio of the length of a side to the sine of its opposite angle is the same for all sides of a triangle.
So, we have:
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ where $a$ , $b$ , and $c$ are the sides opposite angles $A$ , $B$ , and $C$ respectively.

STEP 7

We want to find the distance from Brian to the house, which is side $b$ (opposite angle $B$ ).
We know:
- $a = 180$ meters (distance between Brian and Lashonda) - $\sin A = \sin 49^\circ$ - $\sin B = \sin 73^\circ$

STEP 8

Using the Law of Sines, we set up the equation:
$\frac{180}{\sin 58^\circ} = \frac{b}{\sin 73^\circ}$

STEP 9

Let's solve for $b$ by multiplying both sides by $\sin 73^\circ$ :
$b = \frac{180 \cdot \sin 73^\circ}{\sin 58^\circ}$

STEP 10

Now, let's plug in the values for the sines:
- $\sin 73^\circ \approx 0.9563$ - $\sin 58^\circ \approx 0.8480$

STEP 11

Substitute these into the equation:
$b = \frac{180 \cdot 0.9563}{0.8480}$

STEP 12

Calculate the value:
$b \approx \frac{172.134}{0.8480} \approx 203.0$

STEP 13

The distance from Brian to the house is approximately 203.0 meters.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

What is this asking? We need to find out how far Brian is from a house across the river, using the angles they measured and the distance between Brian and Lashonda. Watch out! Don't forget to use the Law of Sines correctly, and make sure to round your final answer to the nearest tenth!

STEP 2

1. Understand the triangle2. Apply the Law of Sines3. Solve for the distance

STEP 3

Alright, let's picture this: Brian and Lashonda are standing on the riverbank, and the house is across the river.We've got a triangle here, with points AAA, BBB, and CCC.Brian is at AAA, Lashonda is at BBB, and the house is at CCC.The distance between AAA and BBB is **180 meters**.

STEP 4

We know angle AAA is 49∘49^\circ49∘ and angle BBB is 73∘73^\circ73∘.To find angle CCC, we use the fact that the sum of angles in a triangle is 180∘180^\circ180∘.So, we calculate:C=180∘−49∘−73∘C = 180^\circ - 49^\circ - 73^\circC=180∘−49∘−73∘

STEP 5

Plugging in the numbers, we find:C=58∘C = 58^\circC=58∘

STEP 6

STEP 7

We want to find the distance from Brian to the house, which is side bbb (opposite angle BBB).We know:- a=180a = 180a=180 meters (distance between Brian and Lashonda) - sin⁡A=sin⁡49∘\sin A = \sin 49^\circsinA=sin49∘ - sin⁡B=sin⁡73∘\sin B = \sin 73^\circsinB=sin73∘

STEP 8

Using the Law of Sines, we set up the equation:180sin⁡58∘=bsin⁡73∘\frac{180}{\sin 58^\circ} = \frac{b}{\sin 73^\circ}sin58∘180​=sin73∘b​

STEP 9

Let's solve for bbb by multiplying both sides by sin⁡73∘\sin 73^\circsin73∘:b=180⋅sin⁡73∘sin⁡58∘b = \frac{180 \cdot \sin 73^\circ}{\sin 58^\circ}b=sin58∘180⋅sin73∘​

STEP 10

Now, let's plug in the values for the sines:- sin⁡73∘≈0.9563\sin 73^\circ \approx 0.9563sin73∘≈0.9563 - sin⁡58∘≈0.8480\sin 58^\circ \approx 0.8480sin58∘≈0.8480

STEP 11

Substitute these into the equation:b=180⋅0.95630.8480b = \frac{180 \cdot 0.9563}{0.8480}b=0.8480180⋅0.9563​

STEP 12

Calculate the value:b≈172.1340.8480≈203.0b \approx \frac{172.134}{0.8480} \approx 203.0b≈0.8480172.134​≈203.0

STEP 13

The distance from Brian to the house is approximately **203.0 meters**.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Understand the triangle
2. Apply the Law of Sines
3. Solve for the distance

Alright, let's picture this: Brian and Lashonda are standing on the riverbank, and the house is across the river.
We've got a triangle here, with points $A$ , $B$ , and $C$ .
Brian is at $A$ , Lashonda is at $B$ , and the house is at $C$ .
The distance between $A$ and $B$ is 180 meters.

We know angle $A$ is $49^\circ$ and angle $B$ is $73^\circ$ .
To find angle $C$ , we use the fact that the sum of angles in a triangle is $180^\circ$ .
So, we calculate:
$C = 180^\circ - 49^\circ - 73^\circ$

Plugging in the numbers, we find:
$C = 58^\circ$

We want to find the distance from Brian to the house, which is side $b$ (opposite angle $B$ ).
We know:
- $a = 180$ meters (distance between Brian and Lashonda) - $\sin A = \sin 49^\circ$ - $\sin B = \sin 73^\circ$

Using the Law of Sines, we set up the equation:
$\frac{180}{\sin 58^\circ} = \frac{b}{\sin 73^\circ}$

Let's solve for $b$ by multiplying both sides by $\sin 73^\circ$ :
$b = \frac{180 \cdot \sin 73^\circ}{\sin 58^\circ}$

Now, let's plug in the values for the sines:
- $\sin 73^\circ \approx 0.9563$ - $\sin 58^\circ \approx 0.8480$

Substitute these into the equation:
$b = \frac{180 \cdot 0.9563}{0.8480}$

Calculate the value:
$b \approx \frac{172.134}{0.8480} \approx 203.0$

The distance from Brian to the house is approximately 203.0 meters.