ElementarY GeometrYfor Teachers{]<¡arse T}tz<:jx 1ærså*x J

Thomas H. Parker

Professor of Mathematics

Michigan Smte UniversitY

Scon BaldridgeAssistant Professor of Mathematics

Louisiana State UniversitY

July,2007

Copyúght e) 20Oi by Thomas H. Parker and Scott Baldridge'

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Requests for permission to make copies of any parl of the work should be emailed to Thomas

Parker at parker@ math. msu.edu.

Printed and bound by Quebecor World, MI.

Printed in the United States of America.

rsBN 0-9748140-3-2

12ó o CHAPTER 5. AREA

just use this ahitude!

-5. Write T!acher's Solutions to the four problems in problem28 on page 92 of Primary Marh 5A. Make your solutionsclear and complete and at the grade 5 level.

6. Use the area formula for a triangle to obtain the area fbr-mula for a parallelogram by the approach suggested af_ter Exercise 5.3 in this section. Write your answer in the"Teacher's Solution" format: draw and label a picture and

-eive an explanation with equations and a few words.

7.. Do Exercise 3.4 in this section.

8. Here are three methods for finding the area formula of atrapezoid. Show that all three methods lead to the samearea fbrmula. (In each picrure, a and b are the base lensthsand å is the height.)

a) Divide the trapezoid inro two triangles as illustratedbelow. Calculate the area of each triangle and add.

o

Divide the trapezoid in half and pur the top half nexrto tl¡e bottom half. Label and calculate the a¡ea ofthe parallelogram created.

Divjde the trapezoid into two triangles and a rect-angle. Calculate the a¡ea of each and add them to_sether.

(Studv the Textbook!) In NEMI, read Class Activity 3 arthe top of page 334. Answer all the questions. Justifyyour answers to parts (c), (d) and (e).

Using Example 3.6 as a guide, wrire a complereproof for the area formula for a parallelogram basedon the idea that, in the picture below. Area(P) =Area(big rectangle) - 2Area(S).

Any rhombus is a parallelogram, so its area is Base xHeight. But here is second way to calculate its area.

a) We have already seen that the diagonals of a rhom-bus are perpendicular bisectors. Use this fact to showthat the a¡ea of the rhombus is

A(rhombus) =

where c and d are the lengths of the diagonals.

b) lf aABCD is a square (a special rhombus) with sidelen-eth s, then all sides are congruent and all anglesare 90'. In particular, the'diagonals have the samelength (c = d). Ure the formula from part (a) to con-cludethatd= "'12.s.

9.

r0.

I l.

b)

I

7.'o,

c)

10.

I l.

+.

5.

The hypotenuse ofa right triangle is 20 cm and one leg isl2 cm long. Find the area ofthe triangle.

The sides of a triangle are 6 cm, I 3 cm, and l5 cm. Is it a

right triangle?

V/hich of the following sets of numbers could be thelengths of the sides of a righr triangle? In c), k is someunspecified number.

SECTIONó.1 PYTHAGOREANTHEOREM . 13ì

A rhombus has sides I 0 cm long. If one diagonal is I 2 cmlong, how long is the other diagonal?

A teacher put the following problem on a test:

"The longest side of a triangle is 17 cm long and one legis 15 cm long. What is rhe a¡ea of the triangle?"

a) Explain why this problem, as sfated, cannot be an-

swered.

b) Suppose thal the triangle is a right triangle and solvethe problem.

The picture shows a recrangular box with height 3 in.width 4 in. and length 12 in.

a) What is the length otF9lb) How long is the diagond EE? Hint: LEBH is a

right triangle.

4in

President James Garfield discovered a proof of thePythagorean Theorem based on the picture below. In thepicture, the original triangle has been extended to a trape-zoid composed of three triangles.

a) Explain why zl is a right angle.

b) V/hat is the sum ofthe areas ofthe three triangles interms of c, b and c?

c) Using the formula for the area of a trapezoid, whatis the total area in terms of a and b?

d) Prove ¡hat a2 + bz = c2 by equating your answers tob) and c) and simplifying.

a) 3,4,5

d) t6,30,35

b) 6,8, l0

e) 16,30,34

c) 3k,4k,5k

" 1,Í,1

l7 cm.

t1I !.

7. Determine which of the following rriangles are righr tri-angles and name the right angle.

a) In aABC, AB = 8 cm, BC = 9 cm. and AC = 7 cm.

b) ln IPQR, PQ = 15 m. QR = 25 m, and PÃ = 20 m.

c) ln xXYZ, XY = 36 in, YZ = 39 in, and XZ = 15 in.d) In aSIU, SI = 9cm,TU = l5cm. and SU =

20 cm.

e) In tCDE, CD = 8 km, DE = 15 km. and CE =l7 km.

In the figure, DC = 6 cm, CB = l0 cm, AB = 12 cm, andAB ll DC. Find å.

hcm

8.

13.

0 The figure below is an isosceles triangle with legsbase l6 cm, and height å cm.

a) Find the value ofå.b) Find the area of MBC.

tI

Dt

I

D 6cm C

13ó . CHAPTER ó. PYÌHAGOREAN THEOREM WITH APPIICATIONS

a) aif b =3.3andc=8.8,b) bif a=44.4andc=62,c) c if a = 1.24 and b = 3.82.

5. (Calculator) Of the numbers 2542,7814 and ll.88l.which is ralional and which is irrational? Explain vourreasoning.

6. a) One can simplify Vl8 to 3lã. Describe an algo-rithm (list the steps) for similarly simplifying VFwhere N is a whole number. (Your first step shouldbe "find the prime factorizarion of N").

. b) Apply your algorithm to simplify \n508t20-.

7. Give a Teacher's Solution: A window-cleaner has a lad-der which is 5 meters long. He places it so thar ir reachesa windowsill 4 meters from thè ground. How fa¡ fromthe wall is the foot of the ladder? (Begin by drawing adiagram.)

8. V/hich has the shorter diagonal: the square or rhe rectan-gle?

I 1,. l_-l,,trtt4in Sin

Give a Teacher's Solution: A ladder leans against the walland reaches a height of 3 m. If the foot of the ladder is0.8 m from the wall, find, in meters, rhe lengrh of the lad-der. Give your answer correct to I decimal place.

Give a Têacher's Solution: P and Q are on the oppositesides of a pond. M is a point such that PM and QM canbe measured. It is found that PM = 24m, QM = 26mand ISPM = 90". Calculate the distance berween P and

O.

P

24m

(Mental Math) Find a, b, c and d mentally. List the prim-iriue Pythagorean triples that you use along the way.

39m

Calculate the length QR in IPQR.

a*T^A bridge with supportsM# andØ is built across a river.lf AB = 15 m, AC = 4 nt, and AM = M B, find the lengthof the support MC.

This problem teaches students to use algebra to solve a

geometry problem. Give a Teacher Solution. Start by in-troducing a letter for one of the unknown side lengths.

The shortest side of a right triangle is l0 cm. If the differ-ence between the other two sides is 2 cm, find the perime-ter of the triangle.

',,k Here is a "challenger problem" for students who have

solved the routine problems.

A water-plant originally l0 cm above the water surface is

blown 50 cm sideways by a strong wind as shown. Findthe depth ofthe pond.

l0 cm

12.

13.

9.

10.

t4.

15.

4CmI6 cm

20m

/tsm

Pond

tlll.

Vlzq

A * *É irer ¡: :-ocl' oi, Éh * p"v f ål :agfi ¡Ë ¿¡¡-å T! esr¡.en-¡

There are many proofs of the Pythagorean Theorem. Students can benefit from comparingand contrasting several proofs. This section presents a second common proof of the pytha-soreanTheorem' This "windmill proof'' uses a different picrure than the proof in Section 6, but thelogic is very similar.

v/e begin, as before, with a righr triangle wirh legs of length a and b,labeled so that b > a.Again we observe that the sum of the acute angles is 90..

J+_y+90o = l80o.'.Jr+y=90o.

"ûÍ

Ittc Windmill Proof' Draw a segment of length å extending the short leg beyond the right angle.Complete the extended portion of that segment ro a square with sides of length b - a. Extendeach side a distance ¿ to form the "windmill" fi-eure ,horn in the middle. Label the endpointsof the extended segmenrs A, B, c and D and draw quadrilatera l ABCD.

B

D

In this figure, each shaded triangle is congruenr to the originalright angle and legs of length a and b). Consequently,

triangle by SAS (each has a

the four sides of A.BCD eachhave length c, and

each corner angle of A,BCD has measure ,r + y = 90o.

Thus ABCD is a square with sides of rength c, so it has area c2. But then

c- = Area(center E) + 4.Area(original Á)

= (a - b)(a - b) + +.labJ

= a2 -zab+ b2 +2ab.

This simplifies to c2 = a2 + b?. rl

Notice the similarities with the proof given in Section 6.1. Both use SAS to check thattriangles are congruent, both use the fact that the sum of the angles of a triangle is I g0" to checkthat angles are right angles, and both involve computing a¡ea in two diffierent ways.

,ã

'/

Jtr

a-t A

b-aJ

\'C

Caution. Textbook "proofs" of the Pythagorean Theorem often simply display the square-within-square picture above and do the area calculation. They neglect to show how to start witha right triangle and build the figure. They also assume that the cenrer part of the figure is asquare, without explanation. Because some of the logic of the proof is Ieft out, students mayhave questions for the teacher!

The picture below is also common, and confusing. It gives a visual interpretation of theformula az + b? = c2: the three squares have ar"as o2, b2 uid ,r, so the pythagorean Theoremsays that the total area of the two smaller squares is equal to the area of the largest square. Iåispicture is noî a proof of the Pythagorean Theorem, nor is it connected with the way that thePythagorean Theorem is used in applications.

a2+b2 = c2.

Homework Set 22

In the figure below. how long is : if ¡ = 4?r*_r'= l2?

A lawn is in the shape of an equilateral triangle of sideó m. Find the a¡ea of the lawn (Use a calcuìator and givethe answer to 3 significanr digits.)

Suppose that you are given a riangle ABC, with AB = 7 ,BC = ^7 V3, and AC = 14. Vy'hat are the measures of eachangle of the triangle?

In the figure. RQ = 6. Find the following:

4.t. If -t' = 7r If

ô.

.)

lr.pp-o^t. aDEF is a righr angle and rhe measure of angle

E is ó0".

F

a) How long is .i' if : = 8? Ifz = 12? If¡ = ¡2gr

b) How long is x il' :. = 42 If:= l0ß?lf z=u?

c) How long is : if x+-.¡,-¡ = tJ2

R 6fi.

a) RS.

b) sr.C) QT.d) The perimerer of ¡RS 7.e) The area of ¡RS 7.

The dia-eonal of'a square room is l2 m. Find the perime_ter of the room. (Use a calculator and give vour answer rothe nearest meter).

Piqe7. In the figure, CD = 6 cm, BC = 5 cm, and AC = l2cnl.

with a and b marked as shown. Find lensths a and b.

In the figure, AD = 1 cm, DC = 4 cm, and BD = 2 cm.

a) Find the lengths of AB and BC.

b) Show that ¿ABC is a right angle (quote a rheorem).

9. Find the length AC in terms of x (measured in m).

C

l\+'t\¿"¡

10. In the figure below, PS = 20cm, QS = l5cm, 0R =l2 cm, and tQS P = 90'. Find the area and the perimeterof quadrilateral PQRS .

P 20cm S

Each side of a rhombus is 13 feet long and the diagonalis 24 feet long. Sketch and find the length of the otherdiagonal.

12. Draw a rectangle KLMN with KL = 6 m and KN = l0 m.Ma¡k a point P on Ltul such that triangle pNK is isosceleswith equal sides are KP and lfN. Find pN.

13. In the figure, EG = 20 cm. Find the lengths

a) EF,b) DE,

c) FH,

d) GH.

The figure shows two right triangles with their rightgles aligned. Find (a) length c, and (b) angle r.

a) Draw a segment XI of length 12 cm.

b) Extend this to a parallelogram XYZW with YZ =l3 cm and with diagonalfr. perpendicular ro Xf.

c) Find the a¡ea of XYZW.

A hill rises for 100 m at an incline of 45o, then a further60 m at an incline of 30". What is its total heieht /¡?

At one moment, a plane flying horizontally was a distance5000 m vertically above a man. Ten seconds later, it was6000 m from the man. What was the speed of the planein km/h? Draw a sketch and give your answer to 2 signif-icant digits.

11.

t5.

to.

I t.

8.

12 cm

ll.

H