Math / Geometry

QuestionConsider the given triangle. If $b=7$ and $c=25$ then $a=$ $\square$ and the measure of $\angle B=$ degrees. Round your answers to the nearest whole number. $\qquad$ $\qquad$ $\times$ Need a hint?

Studdy Solution

STEP 1

What is this asking? We've got a right triangle, and we know two sides.
We need to find the third side and one of the angles. Watch out! Make sure your calculator is in degree mode if you want your angle in degrees!
Also, remember which sides go with which trig functions.

STEP 2

1. Find side $a$ .
2. Find angle $B$ .

STEP 3

Alright, let's do this!
We've got a right triangle, so we can use the Pythagorean theorem: $a^2 + b^2 = c^2$ .
Remember, $c$ is the hypotenuse, which is the longest side, across from the right angle.

STEP 4

We know $b = \mathbf{7}$ and $c = \mathbf{25}$ , so let's plug those values in: $a^2 + (\mathbf{7})^2 = (\mathbf{25})^2$ .

STEP 5

Now we simplify: $a^2 + \mathbf{49} = \mathbf{625}$ .

STEP 6

To isolate $a^2$ , we subtract $\mathbf{49}$ from both sides: $a^2 = \mathbf{625} - \mathbf{49}$ , so $a^2 = \mathbf{576}$ .

STEP 7

Taking the square root of both sides gives us $a = \sqrt{\mathbf{576}}$ , which means $a = \mathbf{24}$ .
Awesome! We found one side!

STEP 8

Now, let's find the measure of angle $B$ .
We can use the tangent function here.
Remember, $\tan(B) = \frac{\text{opposite}}{\text{adjacent}}$ .

STEP 9

In our triangle, the side opposite angle $B$ is $a = \mathbf{24}$ , and the side adjacent to angle $B$ is $b = \mathbf{7}$ .
So, $\tan(B) = \frac{\mathbf{24}}{\mathbf{7}}$ .

STEP 10

To find angle $B$ , we need to take the inverse tangent (also called arctan) of both sides: $B = \arctan\left(\frac{\mathbf{24}}{\mathbf{7}}\right)$ .

STEP 11

Using a calculator (make sure it's in degree mode!), we get $B \approx \mathbf{73.74}$ degrees.
Rounding to the nearest whole number, we have $B \approx \mathbf{74}$ degrees.
Boom!

STEP 12

We found that $a = \mathbf{24}$ and the measure of angle $B$ is approximately $\mathbf{74}$ degrees.

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QuestionConsider the given triangle. If $b=7$ and $c=25$ then $a=$ $\square$ and the measure of $\angle B=$ degrees. Round your answers to the nearest whole number. $\qquad$ $\qquad$ $\times$ Need a hint?

Studdy Solution

STEP 1

What is this asking? We've got a right triangle, and we know two sides.
We need to find the third side and one of the angles. Watch out! Make sure your calculator is in degree mode if you want your angle in degrees!
Also, remember which sides go with which trig functions.

STEP 2

1. Find side $a$ .
2. Find angle $B$ .

STEP 3

Alright, let's do this!
We've got a right triangle, so we can use the Pythagorean theorem: $a^2 + b^2 = c^2$ .
Remember, $c$ is the hypotenuse, which is the longest side, across from the right angle.

STEP 4

We know $b = \mathbf{7}$ and $c = \mathbf{25}$ , so let's plug those values in: $a^2 + (\mathbf{7})^2 = (\mathbf{25})^2$ .

STEP 5

Now we simplify: $a^2 + \mathbf{49} = \mathbf{625}$ .

STEP 6

To isolate $a^2$ , we subtract $\mathbf{49}$ from both sides: $a^2 = \mathbf{625} - \mathbf{49}$ , so $a^2 = \mathbf{576}$ .

STEP 7

Taking the square root of both sides gives us $a = \sqrt{\mathbf{576}}$ , which means $a = \mathbf{24}$ .
Awesome! We found one side!

STEP 8

Now, let's find the measure of angle $B$ .
We can use the tangent function here.
Remember, $\tan(B) = \frac{\text{opposite}}{\text{adjacent}}$ .

STEP 9

In our triangle, the side opposite angle $B$ is $a = \mathbf{24}$ , and the side adjacent to angle $B$ is $b = \mathbf{7}$ .
So, $\tan(B) = \frac{\mathbf{24}}{\mathbf{7}}$ .

STEP 10

To find angle $B$ , we need to take the inverse tangent (also called arctan) of both sides: $B = \arctan\left(\frac{\mathbf{24}}{\mathbf{7}}\right)$ .

STEP 11

Using a calculator (make sure it's in degree mode!), we get $B \approx \mathbf{73.74}$ degrees.
Rounding to the nearest whole number, we have $B \approx \mathbf{74}$ degrees.
Boom!

STEP 12

We found that $a = \mathbf{24}$ and the measure of angle $B$ is approximately $\mathbf{74}$ degrees.

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QuestionConsider the given triangle. If b=7b=7b=7 and c=25c=25c=25 then a=a=a= □\square□ and the measure of ∠B=\angle B=∠B= degrees. Round your answers to the nearest whole number. \qquad \qquad ×\times× Need a hint?

Studdy Solution

STEP 1

What is this asking? We've got a right triangle, and we know two sides.We need to find the third side and one of the angles. Watch out! Make sure your calculator is in degree mode if you want your angle in degrees!Also, remember which sides go with which trig functions.

STEP 2

1. Find side aaa.2. Find angle BBB.

STEP 3

Alright, let's **do this**!We've got a right triangle, so we can use the **Pythagorean theorem**: a2+b2=c2a^2 + b^2 = c^2a2+b2=c2.Remember, ccc is the **hypotenuse**, which is the longest side, across from the right angle.

STEP 4

We know b=7b = \mathbf{7}b=7 and c=25c = \mathbf{25}c=25, so let's plug those values in: a2+(7)2=(25)2a^2 + (\mathbf{7})^2 = (\mathbf{25})^2a2+(7)2=(25)2.

STEP 5

Now we simplify: a2+49=625a^2 + \mathbf{49} = \mathbf{625}a2+49=625.

STEP 6

To isolate a2a^2a2, we subtract 49\mathbf{49}49 from both sides: a2=625−49a^2 = \mathbf{625} - \mathbf{49}a2=625−49, so a2=576a^2 = \mathbf{576}a2=576.

STEP 7

Taking the square root of both sides gives us a=576a = \sqrt{\mathbf{576}}a=576​, which means a=24a = \mathbf{24}a=24.Awesome! We found one side!

STEP 8

Now, let's find the measure of angle BBB.We can use the **tangent** function here.Remember, tan⁡(B)=oppositeadjacent\tan(B) = \frac{\text{opposite}}{\text{adjacent}}tan(B)=adjacentopposite​.

STEP 9

In our triangle, the side **opposite** angle BBB is a=24a = \mathbf{24}a=24, and the side **adjacent** to angle BBB is b=7b = \mathbf{7}b=7.So, tan⁡(B)=247\tan(B) = \frac{\mathbf{24}}{\mathbf{7}}tan(B)=724​.

STEP 10

To find angle BBB, we need to take the **inverse tangent** (also called **arctan**) of both sides: B=arctan⁡(247)B = \arctan\left(\frac{\mathbf{24}}{\mathbf{7}}\right)B=arctan(724​).

STEP 11

Using a calculator (make sure it's in degree mode!), we get B≈73.74B \approx \mathbf{73.74}B≈73.74 degrees.Rounding to the nearest whole number, we have B≈74B \approx \mathbf{74}B≈74 degrees.Boom!

STEP 12

We found that a=24a = \mathbf{24}a=24 and the measure of angle BBB is approximately 74\mathbf{74}74 degrees.

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

QuestionConsider the given triangle. If $b=7$ and $c=25$ then $a=$ $\square$ and the measure of $\angle B=$ degrees. Round your answers to the nearest whole number. $\qquad$ $\qquad$ $\times$ Need a hint?

What is this asking? We've got a right triangle, and we know two sides.
We need to find the third side and one of the angles. Watch out! Make sure your calculator is in degree mode if you want your angle in degrees!
Also, remember which sides go with which trig functions.

1. Find side $a$ .
2. Find angle $B$ .

Alright, let's do this!
We've got a right triangle, so we can use the Pythagorean theorem: $a^2 + b^2 = c^2$ .
Remember, $c$ is the hypotenuse, which is the longest side, across from the right angle.

We know $b = \mathbf{7}$ and $c = \mathbf{25}$ , so let's plug those values in: $a^2 + (\mathbf{7})^2 = (\mathbf{25})^2$ .

Now we simplify: $a^2 + \mathbf{49} = \mathbf{625}$ .

To isolate $a^2$ , we subtract $\mathbf{49}$ from both sides: $a^2 = \mathbf{625} - \mathbf{49}$ , so $a^2 = \mathbf{576}$ .

Taking the square root of both sides gives us $a = \sqrt{\mathbf{576}}$ , which means $a = \mathbf{24}$ .
Awesome! We found one side!

Now, let's find the measure of angle $B$ .
We can use the tangent function here.
Remember, $\tan(B) = \frac{\text{opposite}}{\text{adjacent}}$ .

In our triangle, the side opposite angle $B$ is $a = \mathbf{24}$ , and the side adjacent to angle $B$ is $b = \mathbf{7}$ .
So, $\tan(B) = \frac{\mathbf{24}}{\mathbf{7}}$ .

To find angle $B$ , we need to take the inverse tangent (also called arctan) of both sides: $B = \arctan\left(\frac{\mathbf{24}}{\mathbf{7}}\right)$ .

Using a calculator (make sure it's in degree mode!), we get $B \approx \mathbf{73.74}$ degrees.
Rounding to the nearest whole number, we have $B \approx \mathbf{74}$ degrees.
Boom!

We found that $a = \mathbf{24}$ and the measure of angle $B$ is approximately $\mathbf{74}$ degrees.