Question

On a coordinate plane, point G is at (2, 8), point H is at (10, 10), point I is at (10, 0), and point J is at (6, negative 4).
Which side has a length of 10 units?
HG
GJ
HI
IJ

Answer 1

To find which side has a length of 10 units, we will calculate the lengths of each segment \( HG \), \( GJ \), \( HI \), and \( IJ \) using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

**1. Length of HG:**

Points \( H(10, 10) \) and \( G(2, 8) \):
\[
HG = \sqrt{(10 - 2)^2 + (10 - 8)^2} = \sqrt{(8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68} \approx 8.25
\]

**2. Length of GJ:**

Points \( G(2, 8) \) and \( J(6, -4) \):
\[
GJ = \sqrt{(6 - 2)^2 + (-4 - 8)^2} = \sqrt{(4)^2 + (-12)^2} = \sqrt{16 + 144} = \sqrt{160} \approx 12.65
\]

**3. Length of HI:**

Points \( H(10, 10) \) and \( I(10, 0) \):
\[
HI = \sqrt{(10 - 10)^2 + (0 - 10)^2} = \sqrt{(0)^2 + (-10)^2} = \sqrt{100} = 10
\]

**4. Length of IJ:**

Points \( I(10, 0) \) and \( J(6, -4) \):
\[
IJ = \sqrt{(6 - 10)^2 + (-4 - 0)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.66
\]

Based on the calculations, the segment with a length of 10 units is **HI**.