Math / Geometry

QuestionWhat is the area, in square units, of triangle BCD? Triangle BCD, with vertices B(-6,-9), C(-2,-4), and D(-8,-3), is drawn inside a rectangle, as shown below.

Studdy Solution

STEP 1

What is this asking? We need to find the area of a triangle inside a square on a coordinate grid. Watch out! Don't forget to use the coordinate values, not just count squares!

STEP 2

1. Identify the coordinates
2. Calculate the square's area
3. Find the triangle's area
4. Subtract to get the shaded area

STEP 3

Alright, let's get our detective hats on and find those sneaky coordinates!
We've got a square with a triangle inside, and we need to figure out where all the points are hiding.

STEP 4

Let's start with the square.
It's sitting pretty from $(-9, -1)$ to $(-1, 7)$ .
Can you see it?
That's our playground for this problem!

STEP 5

Now, let's name our triangle's vertices.
We've got: - Point B at $(-9, -1)$ - Point C at $(-1, -1)$ - Point D at $(-1, 7)$
See how they form a right triangle?
That's going to be super helpful later!

STEP 6

Time to size up our square!
We need to find its area, but first, let's figure out its side length.

STEP 7

The square goes from $-9$ to $-1$ on the x-axis.
That's a difference of: $|-1 - (-9)| = |-1 + 9| = |8| = 8$ So our square is 8 units wide and 8 units tall.
It's a perfect square!

STEP 8

Now, let's calculate the area.
For a square, we multiply the side length by itself: $\text{Area of square} = 8 \cdot 8 = 64 \text{ square units}$

STEP 9

Now for the star of our show - the triangle!
We're going to use the formula for the area of a triangle: $\frac{1}{2} \cdot \text{base} \cdot \text{height}$ .

STEP 10

Let's find the base and height: - The base is the same as the square's width: 8 units - The height is the same as the square's height: 8 units

STEP 11

Now let's plug these into our formula: $\text{Area of triangle} = \frac{1}{2} \cdot 8 \cdot 8 = 32 \text{ square units}$

STEP 12

We're in the home stretch!
The shaded area is what's left when we take the triangle out of the square.

STEP 13

Let's do the math: $\text{Shaded area} = \text{Area of square} - \text{Area of triangle}$ $\text{Shaded area} = 64 - 32 = 32 \text{ square units}$

STEP 14

The area of the shaded region (triangle BCD) is 32 square units.

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QuestionWhat is the area, in square units, of triangle BCD? Triangle BCD, with vertices B(-6,-9), C(-2,-4), and D(-8,-3), is drawn inside a rectangle, as shown below.

Studdy Solution

STEP 1

What is this asking? We need to find the area of a triangle inside a square on a coordinate grid. Watch out! Don't forget to use the coordinate values, not just count squares!

STEP 2

1. Identify the coordinates
2. Calculate the square's area
3. Find the triangle's area
4. Subtract to get the shaded area

STEP 3

Alright, let's get our detective hats on and find those sneaky coordinates!
We've got a square with a triangle inside, and we need to figure out where all the points are hiding.

STEP 4

Let's start with the square.
It's sitting pretty from $(-9, -1)$ to $(-1, 7)$ .
Can you see it?
That's our playground for this problem!

STEP 5

Now, let's name our triangle's vertices.
We've got: - Point B at $(-9, -1)$ - Point C at $(-1, -1)$ - Point D at $(-1, 7)$
See how they form a right triangle?
That's going to be super helpful later!

STEP 6

Time to size up our square!
We need to find its area, but first, let's figure out its side length.

STEP 7

The square goes from $-9$ to $-1$ on the x-axis.
That's a difference of: $|-1 - (-9)| = |-1 + 9| = |8| = 8$ So our square is 8 units wide and 8 units tall.
It's a perfect square!

STEP 8

Now, let's calculate the area.
For a square, we multiply the side length by itself: $\text{Area of square} = 8 \cdot 8 = 64 \text{ square units}$

STEP 9

Now for the star of our show - the triangle!
We're going to use the formula for the area of a triangle: $\frac{1}{2} \cdot \text{base} \cdot \text{height}$ .

STEP 10

Let's find the base and height: - The base is the same as the square's width: 8 units - The height is the same as the square's height: 8 units

STEP 11

Now let's plug these into our formula: $\text{Area of triangle} = \frac{1}{2} \cdot 8 \cdot 8 = 32 \text{ square units}$

STEP 12

We're in the home stretch!
The shaded area is what's left when we take the triangle out of the square.

STEP 13

Let's do the math: $\text{Shaded area} = \text{Area of square} - \text{Area of triangle}$ $\text{Shaded area} = 64 - 32 = 32 \text{ square units}$

STEP 14

The area of the shaded region (triangle BCD) is 32 square units.

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QuestionWhat is the area, in square units, of triangle BCD? Triangle BCD, with vertices B(-6,-9), C(-2,-4), and D(-8,-3), is drawn inside a rectangle, as shown below.

Studdy Solution

STEP 1

What is this asking? We need to find the area of a triangle inside a square on a coordinate grid. Watch out! Don't forget to use the coordinate values, not just count squares!

STEP 2

1. Identify the coordinates2. Calculate the square's area3. Find the triangle's area4. Subtract to get the shaded area

STEP 3

Alright, let's get our detective hats on and find those sneaky coordinates!We've got a square with a triangle inside, and we need to figure out where all the points are hiding.

STEP 4

Let's start with the square.It's sitting pretty from (−9,−1)(-9, -1)(−9,−1) to (−1,7)(-1, 7)(−1,7).Can you see it?That's our playground for this problem!

STEP 5

Now, let's name our triangle's vertices.We've got: - Point B at (−9,−1)(-9, -1)(−9,−1) - Point C at (−1,−1)(-1, -1)(−1,−1) - Point D at (−1,7)(-1, 7)(−1,7)See how they form a right triangle?That's going to be super helpful later!

STEP 6

Time to size up our square!We need to find its area, but first, let's figure out its side length.

STEP 7

The square goes from −9-9−9 to −1-1−1 on the x-axis.That's a difference of: ∣−1−(−9)∣=∣−1+9∣=∣8∣=8|-1 - (-9)| = |-1 + 9| = |8| = 8∣−1−(−9)∣=∣−1+9∣=∣8∣=8 So our square is 8 units wide and 8 units tall.It's a perfect square!

STEP 8

Now, let's calculate the area.For a square, we multiply the side length by itself: Area of square=8⋅8=64 square units\text{Area of square} = 8 \cdot 8 = 64 \text{ square units}Area of square=8⋅8=64 square units

STEP 9

Now for the star of our show - the triangle!We're going to use the formula for the area of a triangle: 12⋅base⋅height\frac{1}{2} \cdot \text{base} \cdot \text{height}21​⋅base⋅height.

STEP 10

Let's find the base and height: - The base is the same as the square's width: 8 units - The height is the same as the square's height: 8 units

STEP 11

Now let's plug these into our formula: Area of triangle=12⋅8⋅8=32 square units\text{Area of triangle} = \frac{1}{2} \cdot 8 \cdot 8 = 32 \text{ square units}Area of triangle=21​⋅8⋅8=32 square units

STEP 12

We're in the home stretch!The shaded area is what's left when we take the triangle out of the square.

STEP 13

STEP 14

The area of the shaded region (triangle BCD) is **32 square units**.

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Start learning now

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

1. Identify the coordinates
2. Calculate the square's area
3. Find the triangle's area
4. Subtract to get the shaded area

Alright, let's get our detective hats on and find those sneaky coordinates!
We've got a square with a triangle inside, and we need to figure out where all the points are hiding.

Let's start with the square.
It's sitting pretty from $(-9, -1)$ to $(-1, 7)$ .
Can you see it?
That's our playground for this problem!

Now, let's name our triangle's vertices.
We've got: - Point B at $(-9, -1)$ - Point C at $(-1, -1)$ - Point D at $(-1, 7)$
See how they form a right triangle?
That's going to be super helpful later!

Time to size up our square!
We need to find its area, but first, let's figure out its side length.

The square goes from $-9$ to $-1$ on the x-axis.
That's a difference of: $|-1 - (-9)| = |-1 + 9| = |8| = 8$ So our square is 8 units wide and 8 units tall.
It's a perfect square!

Now, let's calculate the area.
For a square, we multiply the side length by itself: $\text{Area of square} = 8 \cdot 8 = 64 \text{ square units}$

Now for the star of our show - the triangle!
We're going to use the formula for the area of a triangle: $\frac{1}{2} \cdot \text{base} \cdot \text{height}$ .

Now let's plug these into our formula: $\text{Area of triangle} = \frac{1}{2} \cdot 8 \cdot 8 = 32 \text{ square units}$

We're in the home stretch!
The shaded area is what's left when we take the triangle out of the square.

The area of the shaded region (triangle BCD) is 32 square units.