(B) If $$ \mathrm{A}(4,3), B(7,0), \mathrm{C}(1,-2) $$ and $$ \mathrm{D}(1,2) $$ Prove that : (1) $$ \overline{\mathrm{AD}} / / \overline{\mathrm{BC}} $$ (2) The figure ABCD is trapezoid

Question

UpStudy AI · Accepted Answer

To prove that the line segments $$\overline{AD}$$ and $$\overline{BC}$$ are parallel and that the figure ABCD is a trapezoid, we will first find the slopes of the lines formed by the points $$A$$, $$B$$, $$C$$, and $$D$$.

### Step 1: Calculate the slopes of $$\overline{AD}$$ and $$\overline{BC}$$

1. **Slope of $$\overline{AD}$$**:
   - Points $$A(4, 3)$$ and $$D(1, 2)$$
   - The formula for the slope $$m$$ between two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is:
     $$
     m = \frac{y_2 - y_1}{x_2 - x_1}
     $$
   - Applying this to points $$A$$ and $$D$$:
     $$
     m_{AD} = \frac{2 - 3}{1 - 4} = \frac{-1}{-3} = \frac{1}{3}
     $$

2. **Slope of $$\overline{BC}$$**:
   - Points $$B(7, 0)$$ and $$C(1, -2)$$
   - Applying the slope formula:
     $$
     m_{BC} = \frac{-2 - 0}{1 - 7} = \frac{-2}{-6} = \frac{1}{3}
     $$

### Step 2: Prove that $$\overline{AD} \parallel \overline{BC}$$

Since both slopes are equal:
$$
m_{AD} = m_{BC} = \frac{1}{3}
$$
This implies that the lines $$\overline{AD}$$ and $$\overline{BC}$$ are parallel.

### Step 3: Prove that ABCD is a trapezoid

A trapezoid is defined as a quadrilateral with at least one pair of parallel sides. Since we have shown that $$\overline{AD} \parallel \overline{BC}$$, we can conclude that ABCD is a trapezoid.

### Conclusion

1. We have proven that $$\overline{AD} \parallel \overline{BC}$$.
2. Since ABCD has one pair of parallel sides, it is confirmed that ABCD is a trapezoid.

Thus, both statements are proven:
1. $$ \overline{AD} \parallel \overline{BC} $$
2. The figure ABCD is a trapezoid.
The slopes of $$\overline{AD}$$ and $$\overline{BC}$$ are equal, proving they are parallel. Since ABCD has one pair of parallel sides, it is a trapezoid.

(B) If and
Prove that :
(1)
(2) The figure ABCD is trapezoid

Upstudy AI Solution

Answer

Solution

Step 1: Calculate the slopes of and

Step 2: Prove that

Step 3: Prove that ABCD is a trapezoid

Conclusion

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(B) If and Prove that : (1) (2) The figure ABCD is trapezoid