Question

The coordinates of the vertices of △GHJ are G(−1, 3), H(1, 2), and J(−3, −1).

Drag and drop the choices into each box to correctly complete the sentences.


The slope of GH¯¯¯¯¯¯ is ____________ , the slope of HJ¯¯¯¯¯¯ Is ____________ , the slope of JG¯¯¯¯¯¯ ______________ . △GHJ ____ a right triangle because _________.


A. - 4/3
B. - 1/2
C. 1/2
D. 3/4
E. 2
F. is
G. is not
H. no two of these slopes have a product of -1
I. two of these slopes have a product of -1

Answer 1

To solve this problem, we first need to calculate the slopes of the sides of triangle GHJ using the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

1. **Calculating the slope of GH:**
- Points: G(-1, 3) and H(1, 2)
\[
\text{slope of GH} = \frac{2 - 3}{1 - (-1)} = \frac{-1}{2} = -\frac{1}{2}
\]

2. **Calculating the slope of HJ:**
- Points: H(1, 2) and J(-3, -1)
\[
\text{slope of HJ} = \frac{-1 - 2}{-3 - 1} = \frac{-3}{-4} = \frac{3}{4}
\]

3. **Calculating the slope of JG:**
- Points: J(-3, -1) and G(-1, 3)
\[
\text{slope of JG} = \frac{3 - (-1)}{-1 - (-3)} = \frac{4}{2} = 2
\]

Now we have:
- The slope of \(GH\) is \(-\frac{1}{2}\),
- The slope of \(HJ\) is \(\frac{3}{4}\),
- The slope of \(JG\) is \(2\).

Next, we need to check if triangle GHJ is a right triangle by looking for any pair of slopes that are negative reciprocals (the product should equal -1).

4. **Check pairs of slopes:**
- For GH and HJ:
\[
-\frac{1}{2} \times \frac{3}{4} = -\frac{3}{8} \quad (\text{not } -1)
\]
- For HJ and JG:
\[
\frac{3}{4} \times 2 = \frac{3}{2} \quad (\text{not } -1)
\]
- For GH and JG:
\[
-\frac{1}{2} \times 2 = -1 \quad (\text{they are negative reciprocals})
\]

Since there is a pair of slopes (GH and JG) that are negative reciprocals, triangle GHJ **is** a right triangle.

Now we can fill in the blanks:

The slope of \(GH\) is **B. -1/2**, the slope of \(HJ\) is **D. 3/4**, the slope of \(JG\) is **E. 2**. Triangle \(GHJ\) **F. is** a right triangle because **I. two of these slopes have a product of -1**.

To finalize the completion:

The slope of GH¯¯¯¯¯¯ is **-1/2**, the slope of HJ¯¯¯¯¯¯ is **3/4**, the slope of JG¯¯¯¯¯¯ is **2**. △GHJ **is** a right triangle because **two of these slopes have a product of -1**.