# HP Prime for All

**English**Русский

In this activity, we use the Geometry app to create a quadrilateral. We then create and connect the midpoints of consecutive sides of the quadrilateral to form another quadrilateral and explore the properties of the latter in terms of the former.

- Press
I and tap the Geometry icon. The app opens in its Plot view. - Tap
Cmds , select`4 Polygon`

>`4 Quadrilateral`

. - Tap a location and press
E to select the first vertex of the quadrilateral. Continue to tap and pressE to select the other three vertices. pressJ to deselect the quadrilateral command.*The display now shows a quadrilateral named*`E`

, based on the four points`A`

,`B`

,`C`

, and`D`

. - Tap
Cmds and select`2 Point`

>`3 Midpoint`

. Tap near the midpoint of`AB`

(or tap each of points separately) and pressE . Continue likewise to create the midpoints of the other segments. When you are done, pressJ to deselect the midpoint command. - Repeat Steps 2 and 3 to create quadrilateral
`O`

from points`K`

,`L`

,`M`

, and`N`

.*The display now shows both quadrilaterals. We will now tidy up our construction and making it colorful before beginning our explorations.* - Press
Y to display the Symbolic view of the app. Here, each of the geometric objects is defined symbolically. Objects with checks besides them are displayed in the Plot view. Uncheck each segment by highlighting it and pressing✓ . Scroll down to uncheck the segments`KL`

,`LM`

,`MN`

, and`NK`

as well. - With the segments hidden, return to the Plot view (press
P ) and give the quadrilaterals their own colors. Touch outer quadrilateral and tapOptions >`1 Choose Color`

(orZ >`4 Choose Color`

) and select any color from color picker. - Repeat Step 7 to select dark blue for the inner quadrilateral.
- Press
M to open the Numeric view of the app. Here we define measurements and tests involving our geometric objects. TapNew to start a new measurement. In this case, it is a test. TapCmds to open the menu of commands select`A Tests`

, then select`8 Parallelogram`

. The command is pasted into the command line. Remember that the name of our inner quadrilateral is`GO`

. Type`GO`

between the parentheses and pressE . Tap✓ to check this test for display in the Plot view. - Return to the Plot view to see our constructions and the test result. Drag anywhere outside of the construction to pan the plane so you can see both the test result and the construction.

We are now ready to explore our construction.

- Select one of the vertices of the outer quadrilateral by tapping on it and pressing
E . You can now drag it anywhere within the display with your finger. As you move a vertex, notice that the parallelogram test on`KLMN`

maintains a value of 1, indicating it is always a parallelogram. The parallelogram test can return any of 5 values:`0.`

Not a parallelogram`1.`

A parallelogram only`2.`

A rhombus`3.`

A rectangle`4.`

A square

- Press
Y to return to the Symbolic view. Here you can give the coordinates of points`A`

,`B`

,`C`

, and`D`

exact values. - Select
`GA`

, tap and enter new coordinates`(-3,3)`

. Tap when you are done. - Repeat Step 3 with points
`B`

,`C`

, and`D`

so that you have`A(-3,3)`

,`B(3,3)`

,`C(3,-3)`

and`D(-3,-3)`

, making`ABCD`

a square. - Press
P to return to the Plot view. The display shows the parallelogram test now has a value of 4, indicating that`KLMN`

is a square as well. It seems that`KLMN`

is always at least a parallelogram, no matter where we move the coordinates of points`A`

,`B`

,`C`

, and`D`

(as long as they are not collinear!). To see why this is so, simply construct the diagonals`AC`

and`BD`

. - Tap
Cmds and select`3 Line`

>`1 Segment`

. Tap on point`A`

and pressE . Then tap on point`C`

and pressE . Repeat with points`B`

and`D`

. Then pressJ to deselect the command. Since segment`KL`

joins the midpoints of two sides of`ΔABC`

, then it is parallel to the third side`AC`

; likewise,`MN`

joins the midpoints of two sides of`ΔACD`

, and is thus parallel to`AC`

as well. Thus`KL`

and`MN`

are parallel. The same can be seen for segments`NK`

and`ML`

. - Press
I and tap to save your construction with a name you’ll remember.

With the testing and symbolic abilities of the Geometry app, students can set about the proofs or refutations of conjectures such as these:

- If
`ABCD`

is a rhombus, then`KLMN`

is a rectangle - If
`ABCD`

is a rectangle, then`KLMN`

is a rhombus - If
`ABCD`

is an isosceles trapezoid, then`KLMN`

is a rhombus - If the diagonals of
`ABCD`

are perpendicular, then`KLMN`

is a rectangle - If the diagonals of
`ABCD`

are congruent, then`KLMN`

is a rhombus