## InterA.TriangleMedianRightAngle History

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[]Suitable for 6th-grade on up.

[]Suitable for 6th-grade on up. Can be connected to Thale's Theorem on Circles, Diameters and Right-Triangles where the "longest side" is the diameter and the median is any radius.

- [[GlossaryT/ThalesTheorem|Thales Theorem - Circles, Diameters and Right Triangles]

- [[GlossaryT/ThalesTheorem|Thales Theorem - Circles, Diameters and Right Triangles]

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[]User interacts with construction and notes that the triangle "appears" to be a right triangle. Then, using only the fact that the lower angles of an isosceles triangle are equal, the user "proves" that the angle is in fact 90°. Finally, user constructs the triangle.

[]User interacts with construction and notices that the triangle constructed under the given conditions "appears" to be a right triangle. He then uses standard 6th grade geometric facts to "prove" that the angle is in fact 90°. Finally, in a second interactivity the user constructs the triangle.

[]Understanding triangles and logical conclusions.

[]Understanding triangles, angles and logical conclusions.

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[] Online ActivitiesOnline Activity

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[ valign=middle]Pre-knowledge

[]The sum of the angles of a triangle is 180°, the sum of supplementary angles is 180° the angles at the base of an isosceles triangle are congruent (equal size).

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- What kind of triangle is \Delta BMC ?
- Why does \frac{\alpha}{2}+ \gamma = 90^\circ ?

- You need to construct the solid lines! To construct \overline{CA}
- Click on Attach:GgbActivity/segment_len.jpg Δ, then on point C and then type in "a" (letter a - no quotes) and hit Enter.

- You need to construct the solid lines!
- To construct \overline{CA} , click on Attach:GgbActivity/segment_len.jpg Δ, then on point C and then type in "a" (letter a - no quotes) and hit Enter.

2. Notice that triangle ABC satisfies the theorem no matter slider or point positions. This is because of the construction!

2. Notice that ΔABC satisfies the theorem no matter slider or point positions. This is because of the construction!

- You need to construct the solid lines! To construct \overline{CA}

- Do the same for point A. Then find the intersection point M ...

- Do the same for point A. Then find the intersection point and name it M ...

- Does triangle ABC satisfy the "hypothesis" of the theorem (no matter slider or point positions)? Why do you think this is so?

- Does triangle ABC satisfy the "hypothesis" of the theorem (no matter slider or point positions)?

1. Look at the construction in the **window above** .

- Click the checkbox to show triangle ABC.

Notice that angle <ACB \,\,appears to a right-angle (90°).

- Deselect checkbox to hide triangle ABC

- How many degrees is: \alpha + \beta ?

- Select checkbox: Show Angles.

- How many degrees in: \alpha + \beta ?

- Click and drag point C to move and point A to rotate.

- Click and drag point C to move and point A to rotate.

- Click and drag point C to move and point A to rotate.

- Click and drag point C to move and point A to rotate.

- What is: \alpha + \beta ?

- How many degrees is: \alpha + \beta ?

- What is: 2 \cdot \delta + \beta ?

- How many degrees in: 2 \cdot \delta + \beta ?

- What kind of triangle is \Delta CBM ?

- What kind of triangle is \Delta BMC ?

[l]Theorem: Let \overline{CM}\,\, be the median to the longest side of \overline{AB}\,\, of triangle \Delta ABC .

[l]Theorem: Let \overline{CM}\,\, be the median to the longest side \overline{AB}\,\, of triangle \Delta ABC .

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[l]Theorem: : Let m be the median to the longest side of \overline{AB} of triangle \Delta ABC .

[l]Theorem: Let \overline{CM}\,\, be the median to the longest side of \overline{AB}\,\, of triangle \Delta ABC .

[]User interacts with construction and notes that the triangle "appears" to be a right triangle. Then, using only the fact that the lower angles of an isosceles triangle are equal, the user "proves" that the angle is in fact 90°. Finally, the student constructs the same interactivity.

[]User interacts with construction and notes that the triangle "appears" to be a right triangle. Then, using only the fact that the lower angles of an isosceles triangle are equal, the user "proves" that the angle is in fact 90°. Finally, user constructs the triangle.

- Do the same for point A.
- or complete directions.

- Do the same for point A. Then find the intersection point M ...

[l]Theorem: : Let M be the median to the longest side of \overline{AB} of triangle \Delta ABC .

[l]Theorem: : Let m be the median to the longest side of \overline{AB} of triangle \Delta ABC .

Online Activity

[]Brief

[ valign=middle]Brief

[]Download

[]Requires freeware GeoGebra for offline use.

[ valign=middle]Download

[]Zip for offline use: tri_median_right.zip (includes handout, teachers page, 2 ggb interactivities)

Requires freeware GeoGebra for offline use.

- Does triangle ABC satisfy the "hypothesis" of the theorem (no matter slider or point positions)? Why do you think this is so?

- Does triangle ABC satisfy the "hypothesis" of the theorem (no matter slider or point positions)? Why do you think this is so?

[]User interacts with construction and notes that the triangle "appears" to be a right triangle. Then, using only the fact that the lower angles of an isoceles triangle are equal, the user "proves" that the angle is in fact 90°. Finally, the student constructs the same interactivity.

[]User interacts with construction and notes that the triangle "appears" to be a right triangle. Then, using only the fact that the lower angles of an isosceles triangle are equal, the user "proves" that the angle is in fact 90°. Finally, the student constructs the same interactivity.

[]triangles

[]triangles, medians, right-triangles, construction, interactivity, geogebra

[l]Theorem: A triangle is a right triangle *if and only if* the median to the longest side

(hypotenuse) is exactly half the length of this side.

[l]Theorem: : Let M be the median to the longest side of \overline{AB} of triangle \Delta ABC .

- Click and drag the point C to move and the point A to rotate.

- Click and drag point C to move and point A to rotate.

- Click on Attach:segment_len.jpg Δ, then on point C and then type in "a" (letter a - no quotes) and hit Enter.
- Click on Attach:circle_rad.jpg Δ, then on point C and type in "m".

- Click on Attach:GgbActivity/segment_len.jpg Δ, then on point C and then type in "a" (letter a - no quotes) and hit Enter.
- Click on Attach:GgbActivity/circle_rad.jpg Δ, then on point C and type in "m".

[]Requires freeware GeoGebra for offline use.

[]Requires freeware GeoGebra for offline use.

- What is: \alpha + \beta ?
- What kind of triangle is \triangle CAM
- Why does \delta =\frac{\alpha}{2} ?

- What is: \alpha + \beta ?
- What kind of triangle is \Delta CAM ?
- What is: 2 \cdot \delta + \beta ?
- Why does \delta =\frac{\alpha}{2} ?

- What kind of triangle is \triangle CBM
- Why does \frac{\alpha}{2}+ \gamma = 90^\circ ?

- What kind of triangle is \Delta CBM ?
- Why does \frac{\alpha}{2}+ \gamma = 90^\circ ?

- Click and drag the slider c' to change the length of the median (and the hypotenuse \overline{AB} ).

- Click and drag the slider m to change the length of the median (and the hypotenuse \overline{AB} ).

- What kind of triangle is \triangle CAC'

- What kind of triangle is \triangle CAM

- What kind of triangle is \triangle CBC'

- What kind of triangle is \triangle CBM

- Click on Attach:circle_rad.jpg Δ, then on point C and type in "c'".

- Click on Attach:circle_rad.jpg Δ, then on point C and type in "m".

[]Standard

[]CA 6.MG.2.2, CA Geometry 13.0, ACT PF 24-27

[]Standards

[]CA 6.MG.2.2, CA Geometry 13.0, ACT PF 24-27

- showhide init=hide div=div9 lshow="+" lhide="-"
- )Online Activity Sheet - Meta Data

[]Geometry and Measurement, Geometry

[]Measurement and Geometry, Geometry

Online Activity Sheet

- showhide init=hide div=div9 lshow="+" lhide="-"
- )Online Activity Sheet - Meta Data

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[l]Theorem: A triangle is a right triangle *if and only if* the median to the longest side

(hypotenuse) is exactly half the length of this side.

[]Brief

[]User interacts with construction and notes that the triangle "appears" to be a right triangle. Then, using only the fact that the lower angles of an isoceles triangle are equal, the user "proves" that the angle is in fact 90°. Finally, the student constructs the same interactivity.

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[]Goal

[]Understanding triangles and logical conclusions.

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[]Grade

[]6-9 (6th grade, 7th grade, pre-algebra, geometry)

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[]Strand

[]Geometry and Measurement, Geometry

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[]Standard

[]CA 6.MG.2.2, CA 7.AF.4.2, IS 1.AL.2.4, ACT EE 24-27, ACT GR 24-27

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[]Keywords

[]triangles

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[]Comments

[]Suitable for 6th-grade on up.

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[]Source

[]Linda Fahlberg-Stojanovska (no copyright)

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[]Cost

[]Activity and software is free to use

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[]Download

[]Requires freeware GeoGebra for offline use.

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[]Type

[]Java Applet so requires free sunJava player

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[l]Theorem: A triangle is a right triangle *if and only if* the median to the longest side

(hypotenuse) is exactly half the length of this side.

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- Click on Attach:segment_len.jpg Δ, then on point C and then type in "a" (letter a - no quotes) and hit Enter.
- Click on Attach:circle_rad.jpg Δ, then on point C and type in "c'".

- Click on Attach:ggb/segment_len.jpg Δ, then on point C and then type in "a" (letter a - no quotes) and hit Enter.
- Click on Attach:ggb/circle_rad.jpg Δ, then on point C and type in "c'".
- Do the same for point A.
- or complete directions.

[l]Theorem: A triangle is a right triangle *if and only i*f the median to the longest side

[l]Theorem: A triangle is a right triangle *if and only if* the median to the longest side

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[l width=50%]1. Look at the construction in the **window above**..

2. Now create this construction in the **window below**.

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[l]Theorem: A triangle is a right triangle *if and only i*f the median to the longest side

(hypotenuse) is exactly half the length of this side.

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- Click and drag the slider a to change the length of the side \overline{AC}
- Click and drag the slider c' to change the length of the median (and the hypotenuse \overline{AB} ).

- Click and drag the slider a to change the length of the side \overline{AC}
- Click and drag the slider c' to change the length of the median (and the hypotenuse \overline{AB} ).

Online Activity Sheet

1. A triangle is a right-angle triangle if and only if the median to the hypotenuse (longest side) is exactly half the length of this side.

1. A triangle is a right triangle if and only if the median to the hypotenuse (longest side) is exactly half the length of this side.

Definition: In geometry, the median of a triangle is a line segment joining a vertex and the midpoint of the opposite side.

1. A triangle is a right-angle triangle if and only if the median to the hypotenuse (longest side) is exactly half the length of this side.

1. A triangle is a right-angle triangle if and only if the median to the hypotenuse (longest side) is exactly half the length of this side.

- Click and drag the slider a to change the length of the side \overline{BC}

- Click and drag the slider a to change the length of the side \overline{AC}

- Click and drag the point C to move and the point B to rotate.
- Click the checkbox to show/hide triangle ABC.

- Click and drag the point C to move and the point A to rotate.
- Click the checkbox to show triangle ABC.