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InterMath

Title: Triangle Inside a Rectangle

Problem Statement

A triangle has two shared vertices and one shared side with a rectangle. The third vertex isanywhere on the side opposite of the shared side (see figures above).

How does the area of the triangle compare with the area of the rectangle? Why do you thin

this relationship holds?

Problem setup

I am trying to determine the relationship between the area of a triangle inscribed in the rectangleto the area of the rectangle to the area of the rectangle. Note that the triangle shares a side with

the rectangle and its third vertex lies on the opposite side of the rectangle.

Plans to Solve/Investigate the Problem

Prediction: because, visually, it looks like .First I plan to construct the figure above in GSP. Then I ill !easure the area of thetriangle and !easure the area of the rectangle. I ill use GSP"s !easure!ent featureand the area for!ulas of each shape.

#rea of a triangle e$uals%%#&'() * base * height#rea of a rectangle e$uals%%%#&length * idth

I ill use these areas in order to deter!ine if there is a constant ratio of their areas.+et, I ill begin to !ake a case for hy this is alays true.

Investigation/Exploration of the Problem

In order to discuss ho I ent about solving this proble! I ill do it in a step by stepprocess.

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'. -reate a rectangle: dra a line seg!ent, construct ) lines perpendicular tothe line seg!ent, put a point on one of the perpendicular lines and construct aperpendicular line to that point.

). ighlight to points at a ti!e to !ake sure each side is a seg!ent./. 0easure parallel sides of the rectangle to !ake sure they are e$ual.

1. To construct the triangle inside the rectangle, I placed a point 2 on seg!ent3- and constructed a seg!ent #2 fro! point # to point 2 and constructed aseg!ent 24 fro! point 4 to point 2.

0easure!ents of Triangle 0easure!ents of 5ectangl e

! #2 & 6.'7 c!

! 24 & 8.'9 c!

! 4# & 6.8/ c!

! 4# & 6.8/ c!! -4 & 1.9: c!

! #3 & 1.9: c!

! 3- & 6.8/ c!

4

3 -

#

2

#s you can see in this illustration, the base of the triangle ould be line seg!ent #4.;ine seg!ent #4 is also the length of one side of the rectangle. Therefore the base oftriangle and the length of the rectangle are e$ual.

4

3 -

#

2

! 4#! #3 & 17.1< c!)

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4

3 -

#

! 4#! #3 & 17.1< c!)

. In the above illustration,you can see that the base of the triangle and the length of the rectangle are the sa!e as are theidth of the rectangle and the height of the triangle. The base and height of the triangle then !ust be!ultiplied by to get the area hich is hy the area of the triangle is alays of the rectangle.

Extensions of the Problem

?hat if e etend the sides of the rectangles to lines 3- and #4 rather than line seg!ents 3- and#4. If the verte of the triangle can be anyhere on line 3-, ill the sa!e relationship hold@

g

base

height

4

3 - F2

#

4

3 -

#

2

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The triangle ill still be of the rectangle and the relationship ill still hold. The triangle ill still havethe height of line seg!ent #3 and ill still have the sa!e base of line seg!ent #4. The triangle illhave the sa!e height as ell as the sa!e base.

The triangle has !erely !ade a transfor!ation here the shape has changed. The area of thetriangle ill still hold true to the shape. -onservation also co!es to !ind. #lthough the triangle hasbeen stretched out for lack of better ords, the area still re!ains the sa!e as before.

Author & ontact

Chantel Lewis

GPS Connection: Standard M5M MeasurementM5M!.

". #stimate the area of fundamental geometric plane figures

C. $erive the formula for the area of a triangle %demonstrate and explain its relationship to the area of a

rectangle with the same base and height.&$. 'ind the area of triangles using formulae. Students will extend their understanding of area of

fundamental geometric plane figures.

(( In order to investigate this problem) students must understand geometric plane figures as well as area of

triangles and rectangles.

!in"#s$ to resources% references% lesson plans% and/or other materials