Question

Given triangle JKL on the coordinate plane.
j=(1,7) l=(1,1) k=(6,1)


(4 points).
The length of side JK is Response area

The length of side LK is Response area.

The length of side JL is Response area.

The perimeter of triangle JKL in simplest radical form

Answer 1

To find the lengths of the sides of triangle JKL with vertices \( J(1,7) \), \( K(6,1) \), and \( L(1,1) \), we will use the distance formula, which is given by:

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

### Calculating the Lengths of the Sides:

1. **Length of side JK:**
- \( J(1, 7) \) and \( K(6, 1) \)
\[
JK = \sqrt{(6 - 1)^2 + (1 - 7)^2}
= \sqrt{5^2 + (-6)^2}
= \sqrt{25 + 36}
= \sqrt{61}
\]

2. **Length of side LK:**
- \( L(1, 1) \) and \( K(6, 1) \)
\[
LK = \sqrt{(6 - 1)^2 + (1 - 1)^2}
= \sqrt{5^2 + 0^2}
= \sqrt{25}
= 5
\]

3. **Length of side JL:**
- \( J(1, 7) \) and \( L(1, 1) \)
\[
JL = \sqrt{(1 - 1)^2 + (1 - 7)^2}
= \sqrt{0^2 + (-6)^2}
= \sqrt{36}
= 6
\]

### Summary of the Lengths:
- Length of side JK = \(\sqrt{61}\)
- Length of side LK = \(5\)
- Length of side JL = \(6\)

### Calculating the Perimeter:
The perimeter \( P \) of triangle JKL is calculated by adding the lengths of all sides:
\[
P = JK + LK + JL = \sqrt{61} + 5 + 6 = \sqrt{61} + 11
\]

### Final Answers:
- The length of side JK is \(\sqrt{61}\).
- The length of side LK is \(5\).
- The length of side JL is \(6\).
- The perimeter of triangle JKL in simplest radical form is \(\sqrt{61} + 11\).