Just as an introduction for SDP-partners, this is a theoretical ppt on properties of triangles in which first, 3 properties are formulated and visualised (recalling or introducing new concepts) afterwards, 2 of these properties are proved while building up the proves interactively, pupils draw and write on prefab-sheets which combine multiple slides into 1 page (sheets are included here but not translated)

7.3 Properties of triangles

A C B A middle parallel of a triangle (a line connecting the middles of two sides of the triangle) is // with the third side and has half of its length 1)

A C B Two medians of a triangle (lines through angle and middle of opposite side) divide each other in 2 parts which are in the ratio of 2 to 1 2) 2 1

In a rectangular triangle the height onto the hypothenuse is middleproportional between the line segments 3) in which it divides the hypothenuse A C B h x y h 2 = x. y

Een middenparallel van een driehoek (een lijnstuk dat de ……………………………………………………………… ………………………….. ) is // met de derde zijde en ……………………………. Gegeven:  ABC met M het midden van [AB] en N het midden van [BC] Te bewijzen: MN // AC en …………………... Bewijs: Beschouw  ABC en  MBN : B = ………………….. = ……… (……………..)   ABC …………………… M = A AB wordt door MN en AC gesneden volgens ……………………………….. ……………………………… …………………… |MN| = ………………     A C B 

A middle parallel of a triangle (a line which ………………………………………………………… is // with the third side ……………… Given:  ABC with M the middle of [AB] and N the middle of [BC] To be proved: MN // AC and …………………... A C B connects the middles of two sides of the triangle) and has half of its length M N |MN| = |AC| 1)2)

A C B M N Prove: Consider  ABC and  MBN : = ………………….. in common 2  (……….)  ABC ………….    MBN 1)

A C B M N  ABC ………….   MBN 1)  AB is cut by MN and AC according to ……………………………….. equal corresponding angles  MN // AC

A C B M N  ABC ………….   MBN 2)   = 2 |MN| = |AC|

In een rechthoekige driehoek is de hoogte op de schuine zijde ………….…………………………………. tussen de lijnstukken waarin ze de schuine zijde verdeelt. (zie p A.18) Gegeven: rechthoekige  ABC met BH de hoogetlijn op [AC] Te bewijzen: |BH| 2 = |AH|.|HC| Bewijs: Beschouw  AHB en  BHC : A = …………………………………... C = …………………………………... (……….)   AHB ……………………    A C B ……………………….. of ………………………..

In a rectangular triangle the heigth onto the hypothenuse is ………….………… between the line segments in which it divides the hypothenuse middleproportional A C B Given: rectangular  ABC with BH the perpendicular onto [AC] To be proven: …………….. H |BH| 2 = |AH|.|HC|

A C B H Prove: Consider  AHB and  BHC : Name the angles in B: 1 2  (angle angle)   AHB   BHC   |BH| 2 = |AH|.|HC|

… and now exercises …