Math / Geometry

Question(a) Use the Pythagorean theorem to determine the length of the unknown side of the triangle, (b) determine the perimeter of the triangle, and (c) determine the area of the triangle. The figure is not drawn to scale. a. The length of the unknown side is $\square$ $\square$ (Type a whole number.) b. The perimeter of the triangle is $\square$ $\square$ (Type a whole number.) c. The area of the triangle is $\square$ $\square$ (Type a whole number.)

Studdy Solution

STEP 1

What is this asking? We've got a right triangle, we know two of its sides, and we need to find the third side's length, the triangle's perimeter, and its area! Watch out! Don't mix up the sides!
The hypotenuse is the longest side, opposite the right angle.

STEP 2

1. Find the missing side
2. Calculate the perimeter
3. Calculate the area

STEP 3

Alright, so the Pythagorean theorem tells us that $a^2 + b^2 = c^2$ , where $a$ and $b$ are the legs of the right triangle, and $c$ is the hypotenuse (the longest side).
We know $a = 10$ km and $c = 26$ km, so let's plug those values in!

STEP 4

$10^2 + b^2 = 26^2$

STEP 5

This simplifies to: $100 + b^2 = 676$

STEP 6

To get $b^2$ by itself, we subtract 100 from both sides: $b^2 = 676 - 100$ $b^2 = 576$

STEP 7

Now, take the square root of both sides to find $b$ : $b = \sqrt{576}$ $b = 24$ So, the length of the missing side is 24 km!

STEP 8

The perimeter is the sum of all the sides.
We've got $10$ km, $24$ km, and $26$ km.
Let's add them up! $10 + 24 + 26 = 60$

STEP 9

The perimeter of the triangle is 60 km!

STEP 10

The area of a triangle is $\frac{1}{2} \cdot \text{base} \cdot \text{height}$ .
In a right triangle, the legs are the base and height!

STEP 11

We know the legs are $10$ km and $24$ km.
So, the area is: $\frac{1}{2} \cdot 10 \cdot 24$

STEP 12

$\frac{1}{2} \cdot 240 = 120$

STEP 13

The area of the triangle is 120 km $^2$ !

STEP 14

(a) The length of the unknown side is 24. (b) The perimeter of the triangle is 60. (c) The area of the triangle is 120.

Was this helpful?

Studdy Solution

STEP 1

What is this asking? We've got a right triangle, we know two of its sides, and we need to find the third side's length, the triangle's perimeter, and its area! Watch out! Don't mix up the sides!
The hypotenuse is the longest side, opposite the right angle.

STEP 2

1. Find the missing side
2. Calculate the perimeter
3. Calculate the area

STEP 3

Alright, so the Pythagorean theorem tells us that $a^2 + b^2 = c^2$ , where $a$ and $b$ are the legs of the right triangle, and $c$ is the hypotenuse (the longest side).
We know $a = 10$ km and $c = 26$ km, so let's plug those values in!

STEP 4

$10^2 + b^2 = 26^2$

STEP 5

This simplifies to: $100 + b^2 = 676$

STEP 6

To get $b^2$ by itself, we subtract 100 from both sides: $b^2 = 676 - 100$ $b^2 = 576$

STEP 7

Now, take the square root of both sides to find $b$ : $b = \sqrt{576}$ $b = 24$ So, the length of the missing side is 24 km!

STEP 8

The perimeter is the sum of all the sides.
We've got $10$ km, $24$ km, and $26$ km.
Let's add them up! $10 + 24 + 26 = 60$

STEP 9

The perimeter of the triangle is 60 km!

STEP 10

The area of a triangle is $\frac{1}{2} \cdot \text{base} \cdot \text{height}$ .
In a right triangle, the legs are the base and height!

STEP 11

We know the legs are $10$ km and $24$ km.
So, the area is: $\frac{1}{2} \cdot 10 \cdot 24$

STEP 12

$\frac{1}{2} \cdot 240 = 120$

STEP 13

The area of the triangle is 120 km $^2$ !

STEP 14

(a) The length of the unknown side is 24. (b) The perimeter of the triangle is 60. (c) The area of the triangle is 120.

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've got a right triangle, we know two of its sides, and we need to find the third side's length, the triangle's perimeter, and its area! Watch out! Don't mix up the sides!The hypotenuse is the longest side, opposite the right angle.

STEP 2

1. Find the missing side2. Calculate the perimeter3. Calculate the area

STEP 3

Alright, so the Pythagorean theorem tells us that a2+b2=c2a^2 + b^2 = c^2a2+b2=c2, where aaa and bbb are the legs of the right triangle, and ccc is the **hypotenuse** (the longest side).We know a=10a = 10a=10 km and c=26c = 26c=26 km, so let's plug those values in!

STEP 4

102+b2=26210^2 + b^2 = 26^2102+b2=262

STEP 5

This simplifies to: 100+b2=676100 + b^2 = 676100+b2=676

STEP 6

To get b2b^2b2 by itself, we subtract 100 from both sides: b2=676−100b^2 = 676 - 100b2=676−100 b2=576b^2 = 576b2=576

STEP 7

Now, take the square root of both sides to find bbb: b=576b = \sqrt{576}b=576​ b=24b = 24b=24So, the length of the missing side is **24 km**!

STEP 8

The perimeter is the sum of all the sides.We've got 101010 km, 242424 km, and 262626 km.Let's add them up! 10+24+26=6010 + 24 + 26 = 6010+24+26=60

STEP 9

The perimeter of the triangle is **60 km**!

STEP 10

The area of a triangle is 12⋅base⋅height\frac{1}{2} \cdot \text{base} \cdot \text{height}21​⋅base⋅height.In a right triangle, the legs are the base and height!

STEP 11

We know the legs are 101010 km and 242424 km.So, the area is: 12⋅10⋅24\frac{1}{2} \cdot 10 \cdot 2421​⋅10⋅24

STEP 12

12⋅240=120\frac{1}{2} \cdot 240 = 12021​⋅240=120

STEP 13

The area of the triangle is **120 km2^22**!

STEP 14

(a) The length of the unknown side is **24**. (b) The perimeter of the triangle is **60**. (c) The area of the triangle is **120**.

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.

What is this asking? We've got a right triangle, we know two of its sides, and we need to find the third side's length, the triangle's perimeter, and its area! Watch out! Don't mix up the sides!
The hypotenuse is the longest side, opposite the right angle.

1. Find the missing side
2. Calculate the perimeter
3. Calculate the area

Alright, so the Pythagorean theorem tells us that $a^2 + b^2 = c^2$ , where $a$ and $b$ are the legs of the right triangle, and $c$ is the hypotenuse (the longest side).
We know $a = 10$ km and $c = 26$ km, so let's plug those values in!

$10^2 + b^2 = 26^2$

This simplifies to: $100 + b^2 = 676$

To get $b^2$ by itself, we subtract 100 from both sides: $b^2 = 676 - 100$ $b^2 = 576$

Now, take the square root of both sides to find $b$ : $b = \sqrt{576}$ $b = 24$ So, the length of the missing side is 24 km!

The perimeter is the sum of all the sides.
We've got $10$ km, $24$ km, and $26$ km.
Let's add them up! $10 + 24 + 26 = 60$

The perimeter of the triangle is 60 km!

The area of a triangle is $\frac{1}{2} \cdot \text{base} \cdot \text{height}$ .
In a right triangle, the legs are the base and height!

We know the legs are $10$ km and $24$ km.
So, the area is: $\frac{1}{2} \cdot 10 \cdot 24$

$\frac{1}{2} \cdot 240 = 120$

The area of the triangle is 120 km $^2$ !

(a) The length of the unknown side is 24. (b) The perimeter of the triangle is 60. (c) The area of the triangle is 120.