# Landscape Locations

## Solution

Given:

**Kelsey's garden** is * DEFG*. The coordinates of its vertices are: *D* (^{+}5,^{+}7),
*E* (^{+}9,^{+}7), *F* (^{+}8,^{+}2), *G* (^{+}6,^{+}2).

The other gardens:

**Nadia's garden** is a right triangle, *ABC*. The coordinates of its vertices are:

*A* (^{+}2,^{+}7), *B* (^{+}5,^{+}2), *C* (^{+}2,^{+}2).

**Brandon's garden** is *MNOP*. The coordinates of its vertices are: *M* (^{+}5,^{−}7),
*N* (^{+}9,^{−}7), *O* (^{+}8,^{−}2), *P* (^{+}6,^{−}2).

**Sophia's garden** is a right triangle, *JKL*. The coordinates of its vertices are:

*J* (^{−}2,^{+}7), *K* (^{−}5,^{+}2), *L* (^{−}2,^{+}2).

**Miguel's garden** is a right triangle, *XYZ*. The coordinates of its vertices are:

*X* (^{−}4,^{−}2), *Y* (^{−}1,^{−}7), *Z* (^{−}4,^{−}7).

## Explanation

**Kelsey's garden** is *DEFG*. The coordinates of its vertices are: *D* (^{+}5,^{+}7),
*E* (^{+}9,^{+}7), *F* (^{+}8,^{+}2), *G* (^{+}6,^{+}2).

**Nadia's garden** is a right triangle, *ABC*.

The right angle of her garden is 4 units left of vertex *G* of Kelsey's garden.

Since the vertices for the other angles are given as *A* and *B*, the right angle must be *C*.

Vertex *G* of Kelsey's garden is (^{+}6,^{+}2).

Translate 4 units to the left: (^{+}6,^{+}2) → (^{+}2,^{+}2).

The right angle *C* is (^{+}2,^{+}2).

Vertex *A* of Nadia's garden is up 5 units from the right angle.

Up 5 from (^{+}2,^{+}2) is (^{+}2,^{+}7)

Vertex *B* of Nadia's garden is right 3 units from the right angle.

Right 3 units from (^{+}2,^{+}2) is (^{+}5,^{+}2).

The coordinates of its vertices are:
*A* (^{+}2,^{+}7), *B* (^{+}5,^{+}2), *C* (^{+}2,^{+}2).

**Brandon's garden** is a reflection of Kelsey's garden across the *x*-axis.

It is labeled *MNOP*.

The coordinates of the vertices of Kelsey's garden are
*D* (^{+}5,^{+}7), *E* (^{+}9,^{+}7), *F* (^{+}8,^{+}2), *G* (^{+}6,^{+}2).

To reflect each vertex across the *x*-axis, plot the same *x*-coordinates and the opposite *y*-coordinates:

(^{+}5,^{−}7), (^{+}9,^{−}7), (^{+}8,^{−}2), *G* (^{+}6,^{−}2).

Brandon's garden is *MNOP*. The coordinates of its vertices are:

*M* (^{+}5,^{−}7), *N* (^{+}9,^{−}7), *O* (^{+}8,^{−}2), *P* (^{+}6,^{−}2).

**Sophia's garden** is a reflection of Nadia's garden across the *y*-axis.

It is labeled *JKL*.

The coordinates of Nadia's garden are: *A* (^{+}2,^{+}7), *B* (^{+}5,^{+}2), *C* (^{+}2,^{+}2).

To reflect each vertex across the *y*-axis, plot the opposite *x*-coordinates and the same *y*-coordinates:

(^{−}2,^{+}7), (^{−}5,^{+}2), (^{−}2,^{+}2).

**Sophia's garden** is *JKL*. Its coordinates are *J* (^{−}2,^{+}7), *K* (^{−}5,^{+}2), *L* (^{−}2,^{+}2).

**Miguel's garden** is a translation of Nadia's garden 6 units to the left and 9 units down. It is labeled *XYZ*.

Begin with the vertices of Nadia's garden: *A* (^{+}2,^{+}7), *B* (^{+}5,^{+}2), *C* (^{+}2,^{+}2).

Translate each 6 units to the left and 9 units down.

*A* (^{+}2,^{+}7) → (^{-}4,^{−}2)

*B* (^{+}5,^{+}2) → (^{-}1,^{-}7)

*C* (^{+}2,^{+}2) → (^{-}4,^{-}7)

Miguel's garden is a right triangle, *XYZ*. The coordinates of its vertices are:
*X* (^{-}4,^{-}2), *Y* (^{-}1,^{-}7), *Z* (^{-}4,^{-}7).