Pythagorean Theorem

Finding Distances in Right Triangles

Lesson Overview

Instructor Presentation

Introduction

Following is the definition of the Pythagorean Theorem:

(a^2)+(b^2) = (c^2)

True only for right triangles

Goals

Students will be able to identify cases where the Pythagorean Theorem applies, and calculate the distance of each side based on various unknowns.

Students will apply this knowledge to story problems that outline real-world situations for conceptual understanding.

Concept Exploration

The 3-4-5 Triangle

The 3-4-5 triangle is a case in which the distances of each side of the triangle are integers.  Let's apply the Pythagorean Theorem and make sure this is true:

Manipulate[Labeled[Graphics[{RGBColor[1,.95,.75],Polygon[{{0,0},{a,0},{a,b}}],Thickness[.01],PointSize[.02],RGBColor[.67,.75,.15],Line[{{0,0},{a,b}}],Point[Table[k/Sqrt[a^2+b^2]{a,b},{k,0,Floor[Sqrt[a^2+b^2]]}]],RGBColor[1,.26,0],Line[{{0,0},{a,0}}],Point[Table[{i,0},{i,0,a}]],RGBColor[.12,.61,.78],Line[{{a,0},{a,b}}],Point[Table[{a,j},{j,0,b}]],Black,Thickness[.005],Inset[Graphics@Line[{{0,0},{0,.8},{.8,.8}}],{a,0},{.8,0},{.5,.5}],Text[Style["a = "<>ToString[NumberForm[N[a],{4,2},NumberPadding->{"","0"}]],12, Large],{a/2,-.75}],Text[Style["b = "<>ToString[NumberForm[N[b],{4,2},NumberPadding->{"","0"}]],12, Large],{a+.5,b/2},{-1,0}],Text[Style["c = "<>ToString[NumberForm[N[Sqrt[a^2+b^2]],{5,2},NumberPadding->{"","0"}]],12, Large],{a/2-.5,b/2+.5},{1,0}]},ImageSize->{478,335},Axes->False,PlotRange->{{-5,15},{-2,10.5}},AspectRatio->Automatic],TraditionalForm@Text@Style[Grid[{{"(a^2)","+","(b^2)","=","(c^2)"},{Superscript[NumberForm[a,{4,2},NumberPadding->{" ","0"}],2],"+",Superscript[NumberForm[b,{4,2},NumberPadding->{" ","0"}],2],"=",Superscript[NumberForm[N[Sqrt[a^2+b^2]],{5,2},NumberPadding->{" ","0"}],2]},{NumberForm[a^2,{4,2},NumberPadding->{" ","0"}],"+",NumberForm[b^2,{4,2},NumberPadding->{" ","0"}],"=",NumberForm[N[a^2+b^2],{5,2},NumberPadding->{"  ","0"}]}},Alignment->{Right,Baseline},ItemSize->{{5,2.2,4.5,2.2,6},1.5}],12],Top],{{a,3,"a"},.01,10},{{b,4,"b"},.01,10}]

Explore Other Cases

Let's look at other distances, starting with a = 7 and b = 6 where the distances might not be integers:

Manipulate[Labeled[Graphics[{RGBColor[1,.95,.75],Polygon[{{0,0},{a,0},{a,b}}],Thickness[.01],PointSize[.02],RGBColor[.67,.75,.15],Line[{{0,0},{a,b}}],Point[Table[k/Sqrt[a^2+b^2]{a,b},{k,0,Floor[Sqrt[a^2+b^2]]}]],RGBColor[1,.26,0],Line[{{0,0},{a,0}}],Point[Table[{i,0},{i,0,a}]],RGBColor[.12,.61,.78],Line[{{a,0},{a,b}}],Point[Table[{a,j},{j,0,b}]],Black,Thickness[.005],Inset[Graphics@Line[{{0,0},{0,.8},{.8,.8}}],{a,0},{.8,0},{.5,.5}],Text[Style["a = "<>ToString[NumberForm[N[a],{4,2},NumberPadding->{"","0"}]],12, Large],{a/2,-.75}],Text[Style["b = "<>ToString[NumberForm[N[b],{4,2},NumberPadding->{"","0"}]],12, Large],{a+.5,b/2},{-1,0}],Text[Style["c = "<>ToString[NumberForm[N[Sqrt[a^2+b^2]],{5,2},NumberPadding->{"","0"}]],12, Large],{a/2-.5,b/2+.5},{1,0}]},ImageSize->{478,335},Axes->False,PlotRange->{{-5,15},{-2,10.5}},AspectRatio->Automatic],TraditionalForm@Text@Style[Grid[{{"(a^2)","+","(b^2)","=","(c^2)"},{Superscript[NumberForm[a,{4,2},NumberPadding->{" ","0"}],2],"+",Superscript[NumberForm[b,{4,2},NumberPadding->{" ","0"}],2],"=",Superscript[NumberForm[N[Sqrt[a^2+b^2]],{5,2},NumberPadding->{" ","0"}],2]},{NumberForm[a^2,{4,2},NumberPadding->{" ","0"}],"+",NumberForm[b^2,{4,2},NumberPadding->{" ","0"}],"=",NumberForm[N[a^2+b^2],{5,2},NumberPadding->{"  ","0"}]}},Alignment->{Right,Baseline},ItemSize->{{5,2.2,4.5,2.2,6},1.5}],12],Top],{{a,8,"a"},.01,10},{{b,8,"b"},.01,10}]

We can explore many other specific cases with the sliders above

Examples

Example 1

Manipulate[
Mouseover[
Labeled[Graphics[{RGBColor[1,.95,.75],Polygon[{{0,0},{a,0},{a,b}}],Thickness[.01],PointSize[.02],RGBColor[.67,.75,.15],Line[{{0,0},{a,b}}],Point[Table[k/Sqrt[a^2+b^2]{a,b},{k,0,Floor[Sqrt[a^2+b^2]]}]],RGBColor[1,.26,0],Line[{{0,0},{a,0}}],Point[Table[{i,0},{i,0,a}]],RGBColor[.12,.61,.78],Line[{{a,0},{a,b}}],Point[Table[{a,j},{j,0,b}]],Black,Thickness[.005],Inset[Graphics@Line[{{0,0},{0,.8},{.8,.8}}],{a,0},{.8,0},{.5,.5}],Text[Style["a = "<>ToString[NumberForm[N[a],{4,2},NumberPadding->{"","0"}]],12, Large],{a/2,-.75}],Text[Style["b = "<>ToString[NumberForm[N[b],{4,2},NumberPadding->{"","0"}]],12, Large],{a+.5,b/2},{-1,0}],Text[Style["c",12, Large],{a/2-.5,b/2+.5},{1,0}]},ImageSize->{478,335},Axes->False,PlotRange->{{-5,15},{-2,10.5}},AspectRatio->Automatic],TraditionalForm@Text@Style["...mouseover for answer"],Top],
Labeled[Graphics[{RGBColor[1,.95,.75],Polygon[{{0,0},{a,0},{a,b}}],Thickness[.01],PointSize[.02],RGBColor[.67,.75,.15],Line[{{0,0},{a,b}}],Point[Table[k/Sqrt[a^2+b^2]{a,b},{k,0,Floor[Sqrt[a^2+b^2]]}]],RGBColor[1,.26,0],Line[{{0,0},{a,0}}],Point[Table[{i,0},{i,0,a}]],RGBColor[.12,.61,.78],Line[{{a,0},{a,b}}],Point[Table[{a,j},{j,0,b}]],Black,Thickness[.005],Inset[Graphics@Line[{{0,0},{0,.8},{.8,.8}}],{a,0},{.8,0},{.5,.5}],Text[Style["a = "<>ToString[NumberForm[N[a],{4,2},NumberPadding->{"","0"}]],12, Large],{a/2,-.75}],Text[Style["b = "<>ToString[NumberForm[N[b],{4,2},NumberPadding->{"","0"}]],12, Large],{a+.5,b/2},{-1,0}],Text[Style["c = "<>ToString[NumberForm[N[Sqrt[a^2+b^2]],{5,2},NumberPadding->{"","0"}]],12, Large],{a/2-.5,b/2+.5},{1,0}]},ImageSize->{478,335},Axes->False,PlotRange->{{-5,15},{-2,10.5}},AspectRatio->Automatic],TraditionalForm@Text@Style[Grid[{{"(a^2)","+","(b^2)","=","(c^2)"},{Superscript[NumberForm[a,{4,2},NumberPadding->{" ","0"}],2],"+",Superscript[NumberForm[b,{4,2},NumberPadding->{" ","0"}],2],"=",Superscript[NumberForm[N[Sqrt[a^2+b^2]],{5,2},NumberPadding->{" ","0"}],2]},{NumberForm[a^2,{4,2},NumberPadding->{" ","0"}],"+",NumberForm[b^2,{4,2},NumberPadding->{" ","0"}],"=",NumberForm[N[a^2+b^2],{5,2},NumberPadding->{"  ","0"}]}},Alignment->{Right,Baseline},ItemSize->{{5,2.2,4.5,2.2,6},1.5}],12],Top]
],
{{a,8,"a"},.01,10},{{b,8,"b"},.01,10}]

Use sliders above to demonstrate:

(1)  Assume, a = 4 and b = 6, what is the distance of c?

(2)  Assume, a = 8 and b = 8, what is the distance of c?

(3)  Assume, a = 6 and b = 4, what is the distance of c?

(4)  Should (3) be the same as distances a = 4 and b = 6?

Examples

Example 2

Manipulate[
Mouseover[
Labeled[Graphics[{RGBColor[1,.95,.75],Polygon[{{0,0},{a,0},{a,(Sqrt[(c^2)-(a^2)])}}],Thickness[.01],PointSize[.02],RGBColor[.67,.75,.15],Line[{{0,0},{a,(Sqrt[(c^2)-(a^2)])}}],Point[Table[k/Sqrt[a^2+((c^2)-(a^2))]{a,(Sqrt[(c^2)-(a^2)])},{k,0,Floor[Sqrt[a^2+((c^2)-(a^2))]]}]],RGBColor[1,.26,0],Line[{{0,0},{a,0}}],Point[Table[{i,0},{i,0,a}]],RGBColor[.12,.61,.78],Line[{{a,0},{a,(Sqrt[(c^2)-(a^2)])}}],Point[Table[{a,j},{j,0,(Sqrt[(c^2)-(a^2)])}]],Black,Thickness[.005],Inset[Graphics@Line[{{0,0},{0,.8},{.8,.8}}],{a,0},{.8,0},{.5,.5}],Text[Style["a = "<>ToString[NumberForm[N[a],{4,2},NumberPadding->{"","0"}]],12, Large],{a/2,-.75}],Text[Style["b",12, Large],{a+.5,((Sqrt[(c^2)-(a^2)])/2)},{-1,0}],Text[Style["c = "<>ToString[NumberForm[N[c],{5,2},NumberPadding->{"","0"}]],12, Large],{a/2-.5,((Sqrt[(c^2)-(a^2)])/2)+.5},{1,0}]},ImageSize->{478,335},Axes->False,PlotRange->{{-5,15},{-2,10.5}},AspectRatio->Automatic],TraditionalForm@Text@Style["...mouseover for answer",12, Large],Top],
Labeled[Graphics[{RGBColor[1,.95,.75],Polygon[{{0,0},{a,0},{a,(Sqrt[(c^2)-(a^2)])}}],Thickness[.01],PointSize[.02],RGBColor[.67,.75,.15],Line[{{0,0},{a,(Sqrt[(c^2)-(a^2)])}}],Point[Table[k/Sqrt[a^2+((c^2)-(a^2))]{a,(Sqrt[(c^2)-(a^2)])},{k,0,Floor[Sqrt[a^2+((c^2)-(a^2))]]}]],RGBColor[1,.26,0],Line[{{0,0},{a,0}}],Point[Table[{i,0},{i,0,a}]],RGBColor[.12,.61,.78],Line[{{a,0},{a,(Sqrt[(c^2)-(a^2)])}}],Point[Table[{a,j},{j,0,(Sqrt[(c^2)-(a^2)])}]],Black,Thickness[.005],Inset[Graphics@Line[{{0,0},{0,.8},{.8,.8}}],{a,0},{.8,0},{.5,.5}],Text[Style["a = "<>ToString[NumberForm[N[a],{4,2},NumberPadding->{"","0"}]],12, Large],{a/2,-.75}],Text[Style["b = "<>ToString[NumberForm[N[(Sqrt[(c^2)-(a^2)])],{4,2},NumberPadding->{"","0"}]],12, Large],{a+.5,((Sqrt[(c^2)-(a^2)])/2)},{-1,0}],Text[Style["c = "<>ToString[NumberForm[N[c],{5,2},NumberPadding->{"","0"}]],12, Large],{a/2-.5,((Sqrt[(c^2)-(a^2)])/2)+.5},{1,0}]},ImageSize->{478,335},Axes->False,PlotRange->{{-5,15},{-2,10.5}},AspectRatio->Automatic],TraditionalForm@Text@Style["(a^2) + (b^2) = (c^2)",12, Large],Top]
],
{{a,6,"a"},.01,8},{{c,8,"c"},8,10}]

Use sliders above to demonstrate:

(1)  Assume, a = 3 and c = 8, what is the distance of b?

(2)  Assume, a = 8 and c = 8, what is the distance of b?

(3)  Does the answer to (2) make sense?  Why?

Examples

Example 3

Use the Solve command in Mathematica to correctly set up the equation and solve for the unknown value.

For example, if a right triangle has distances where a = 5.25 and c = 7.15, you can use the Solve command in Mathematica to calculate the value of b:

Solve[(5.25^2)+(b^2)==(7.15^2), b]

We're interested in the positive value b = 4.85386

Problems:

(1)  Amy is thinking about buying a new television and wants to figure out if it will fit in her entertainment center.  A 52-inch television is 23 inches tall (and 52 inches diagonal distance).  How wide is the television?

Solve[lhs==rhs,var]

(2)  Amy, the television shopper, notices that wide-screen televisions are twice as wide as they are tall.  What is the height and width of a 47-inch television (47 inches diagonal distance)?

Solve[lhs==rhs,var]

(3)  Tom is standing at the edge of a lake looking at his cabin.  He wonders if it will be faster to drive his boat back to his cabin, or drive his car down two roads around the lake.  He has a digital measuring device, and measures the distance across the lake as 3.2 miles in a north-east direction.  The two roads travel due east, then due north, and he measures the first road going due east at 1.25 miles.

What is the distance of the road that travels due north?

Solve[lhs==rhs,var]

(4)  Extra Credit:  Assuming that Tom can travel at 20 miles per hour on his older boat, and 30 miles per hour by car on the two roads, which is the faster mode of transportation?

Solve[lhs==rhs,var]

Examples Answer Key

Example 1

(1)  Assume, a = 4 and b = 6, what is the distance of c?

7.21

(2)  Assume, a = 8 and b = 8, what is the distance of c?

11.31

(3)  Assume, a = 6 and b = 4, what is the distance of c?

7.21

(4)  Should (3) be the same as distances a = 4 and b = 6?

Yes, the triangles are mirror images of each other.

Example 2

(1)  Assume, a = 3 and c = 8, what is the distance of b?

7.41

(2)  Assume, a = 8 and c = 8, what is the distance of b?

0

(3)  Does the answer to (2) make sense?  Why?

Yes, with these dimensions, the shape ends up being two lines (one on top of the other) rather than a triangle.

Example 3

Problems:

(1)  Amy is thinking about buying a new television and wants to figure out if it will fit in her entertainment center.  A 52-inch television is 23 inches tall (and 52 inches diagonal distance).  How wide is the television?

Solve[(23.0^2)+(b^2)==(52.0^2),b]

46.64 inches

(2)  Amy, the television shopper, notices that wide-screen televisions are twice as wide as they are tall.  What is the height and width of a 47-inch television (47 inches diagonal distance)?

Solve[{(a^2)+(b^2)==(47.0^2), a==b},{a,b}]

Height and width are both 33.23 inches

(3)  Tom is standing at the edge of a lake looking at his cabin.  He wonders if it will be faster to drive his boat back to his cabin, or drive his car down two roads around the lake.  He has a digital measuring device, and measures the distance across the lake as 3.2 miles in a north-east direction.  The two roads travel due east, then due north, and he measures the first road going due east at 1.25 miles.

What is the distance of the road that travels due north?

Solve[(1.25^2)+(b^2)==(3.2^2),b]

2.95 miles

(4)  Extra Credit:  Assuming that Tom can travel at 20 miles per hour on his older boat, and 30 miles per hour by car on the two roads, which is the faster mode of transportation?

distance = rate * time
Boat:
3.2 miles = 20 miles/hour * t
Car:
(2.95 miles + 1.25 miles) = 30 miles/hour * t

Solve[3.2 miles == 20 miles/hour*t, t]
Solve[(2.95 miles + 1.25 miles)==30 miles/hour*t, t]

It's very close, but the travel time by car is shorter than the travel time by boat, even though the distance by boat is shorter.