Question

The area of triangle ABD is 1/4 the area of triangle ABC. Find y, the missing coordinate of vertex D.

A(0,0) B(8.2,0) C(4.6, 4), D(3,y)

Y=1
Y=4.1
Y=25.6
Y=1.6

Answer 1

Are you sure

Answer 2

To find the missing coordinate \( y \) of vertex \( D(3, y) \), we first need to calculate the area of triangle \( ABC \) and then determine the coordinates of \( D \) such that the area of triangle \( ABD \) is \( \frac{1}{4} \) of the area of triangle \( ABC \).

### Step 1: Calculate the area of triangle \( ABC \)

The area of a triangle with vertices at \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:

\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]

Using the coordinates of points:
- \( A(0, 0) \)
- \( B(8.2, 0) \)
- \( C(4.6, 4) \)

we can find the area of triangle \( ABC \):

\[
\text{Area}_{ABC} = \frac{1}{2} \left| 0(0 - 4) + 8.2(4 - 0) + 4.6(0 - 0) \right|
\]

Calculating step by step:

\[
= \frac{1}{2} \left| 0 + 8.2 \cdot 4 + 0 \right| = \frac{1}{2} \left| 32.8 \right| = \frac{32.8}{2} = 16.4
\]

### Step 2: Find the area of triangle \( ABD \)

We need the area of triangle \( ABD \) to be \( \frac{1}{4} \) of the area of triangle \( ABC \):

\[
\text{Area}_{ABD} = \frac{1}{4} \times 16.4 = 4.1
\]

### Step 3: Calculate the area of triangle \( ABD \)

Using the coordinates of points \( A(0, 0) \), \( B(8.2, 0) \), and \( D(3, y) \), we calculate the area of triangle \( ABD \):

\[
\text{Area}_{ABD} = \frac{1}{2} \left| 0(0 - y) + 8.2(y - 0) + 3(0 - 0) \right|
\]

This simplifies to:

\[
= \frac{1}{2} \left| 8.2y \right| = 4.1y
\]

### Step 4: Set up the equation

We set the area of triangle \( ABD \) equal to \( 4.1 \):

\[
4.1y = 4.1
\]

### Step 5: Solve for \( y \)

Dividing both sides by \( 4.1 \):

\[
y = 1
\]

### Conclusion

Thus, the missing coordinate \( y \) of vertex \( D \) is

\[
\boxed{1}
\]

Answer 3

Let's go through the calculations step by step to ensure accuracy in determining the area of triangles \( ABC \) and \( ABD \):

### Step 1: Calculate the area of triangle \( ABC \)

Using the vertices \( A(0, 0) \), \( B(8.2, 0) \), and \( C(4.6, 4) \):

\[
\text{Area}_{ABC} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]

Substituting in the points:

\[
\text{Area}_{ABC} = \frac{1}{2} \left| 0(0 - 4) + 8.2(4 - 0) + 4.6(0 - 0) \right|
\]
\[
= \frac{1}{2} \left| 0 + 8.2 \cdot 4 + 0 \right| = \frac{1}{2} \left| 32.8 \right| = \frac{32.8}{2} = 16.4
\]

### Step 2: Find the area of triangle \( ABD \)

We desire the area of triangle \( ABD \) to be \( \frac{1}{4} \times 16.4 = 4.1 \).

**Using the vertices \( A(0, 0) \), \( B(8.2, 0) \), and \( D(3, y) \)**:

\[
\text{Area}_{ABD} = \frac{1}{2} \left| 0(0 - y) + 8.2(y - 0) + 3(0 - 0) \right|
\]
\[
= \frac{1}{2} \left| 8.2y \right| = 4.1y
\]

### Step 3: Set the area equal to \( 4.1 \)

Setting up the equation:

\[
4.1y = 4.1
\]

### Step 4: Solve for \( y \)

Dividing both sides by \( 4.1 \):

\[
y = 1
\]

### Conclusion

After verifying all calculations carefully, the answer is indeed \( y = 1 \). Therefore, the missing coordinate \( y \) of vertex \( D \) is \( \boxed{1} \).

If you have specific concerns or areas you would like me to clarify further, please let me know!