Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

RO-Q3-MATH9-Module-4-with-Answer-Key.pdf, Study notes of Mathematics

Mariano Marcos State University (MMSU)Mathematics

The module contains only one lesson: Lesson 1: Solve problems involving parallelograms, trapezoids and kites. After going through this module you are expected ...

Typology: Study notes

2021/2022
On special offer
30 Points
Discount

Limited-time offer

Uploaded on 08/01/2022

hal_s95
hal_s95 🇵🇭

4.4

(655)

10K documents

1 / 17

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
1
Z
est for
P
rogress
Z
eal of
P
artnership
9
Republic of the Philippines
Department of Education
Mathematics
Quarter 3 - Module 4:
Word Problems: Parallelogram, Trapezoid
and Kite
Name of Learner: ___________________________
Grade & Section: ___________________________
Name of School: ___________________________
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
Discount

On special offer

Partial preview of the text

Download RO-Q3-MATH9-Module-4-with-Answer-Key.pdf and more Study notes Mathematics in PDF only on Docsity!

1

Z est for P rogress

Z eal of P artnership

Republic of the Philippines

Department of Education

Mathematics

Quarter 3 - Module 4:

Word Problems: Parallelogram, Trapezoid

and Kite

Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

What I Need to Know

The module contains only one lesson: Lesson 1: Solve problems involving parallelograms, trapezoids and kites.

After going through this module you are expected to:

  1. illustrate and solve problems involving parallelograms, trapezoids and kites; and
  2. solve problems involving parallelograms, trapezoids and kites.

What I Know

Directions: Encircle the letter of the correct answer.

  1. How many pair of base angle has isosceles trapezoid? A. 1 B. 2 C. 3 D. 4
  2. In a parallelogram LEND, LE and ND are parallel sides. Which of the following properties of a parallelogram can be used to find the length of the parallel side? A. In a parallelogram, any two opposite angles are congruent. B. In a parallelogram, any two opposite sides are congruent. C. In a parallelogram, any two consecutives angles are supplementary D. The diagonals of a parallelogram bisect each other.
  3. FAME is a parallelogram with angle F measures 40^0 what will be the measure of its opposite angle? A. 1400 B. 50^0 C. 40^0 D. 30^0
  4. In an isosceles trapezoid CARE, measure of angle C is 50^0

What is the measure of angle E?

A. 130^0 B. 110^0 C. 60^0 D. 50^0

  1. The diagonals of the kite RIDE have the measure of 50cm and 15cm, What is the area of the kite? A. 300cm^2 B. 325cm^2 C. 350cm^2 D. 375cm^2
  2. A quadrilateral MAID is a parallelogram.MA =(2x+3) cm and ID=(x + 5)cm ,

What is the measure of the two sides? A. 1cm B. 3cm C. 5cm D. 7cm

What’s New

Activity 2: True or False! Directions : Write T if the statement is correct and if the statement is false write the correct word to make the statement correct. _________1. In a parallelogram, any two opposites sides are congruent (≅).

_________2. In a parallelogram, any two consecutive angles are congruent (≅). _________3. In a parallelogram, any two consecutive angles are complementary. _________4. A diagonal of a parallelogram from two congruent triangle. _________5. The median of a trapezoid is parallel (‖) to each base and its length is one third the sum of the lengths of the bases. _________6. The base angles of an isosceles trapezoid are congruent (≅). _________7. Opposite angles of an isosceles trapezoids are supplementary. _________8. The diagonals of an isosceles trapezoid are congruent (≅). _________9. The diagonals of a kite are perpendicular to each other. ________10. The area of a kite is half the product of the lengths of its diagonal.

What is it

Solving Problems involving Parallelograms, Trapezoids and Kites

The figure below shows the different kinds of quadrilateral. Each kind has its own properties.

QUADRILATERAL

Parallelogram Trapezoid

Rectangle Square Rhombus

Kite

 Let us recall on the properties of a parallelogram Properties of a parallelogram.

  1. In a parallelogram, any two opposites sides are congruent (≅).
  2. In a parallelogram , any two opposite angles are congruent(≅).
  3. In a parallelogram , any two consecutive angles are supplementary.
  4. The diagonals of a parallelogram bisect each other.
  5. A diagonal of a parallelogram form two congruent triangle.

These properties would help us solve problems involving parallelograms.

Let us try a look at this given example 1.

Given a parallelogram CAVE, CA=(x+6) cm and EV=(4x-15) cm CE=(2y+1) cm and AV= (y+3)cm.

  1. How long is CA?
  2. Is CA ≅ EV?
  3. How long is AV?

Solution:

Since CA and EV, EC and AV are sides which are parallel and opposite to each other then

we can apply property number 1 which states that in a parallelogram opposite sides are congruent.

  1. How long is CA?

Given: CA=(x+6)cm EV=(4x-15)cm CA≅EV In a parallelogram , opposite sides are congruent. x + 6 = 4x – 15 By substitution x + 6 – 6 = 4x – 15 – 6 Addition Property of Equality x - 4x = 4x – 4x - 21 Addition Property of Equality -3x = - -3x = - -3 - x = 7 value of x

By substitution, solve for CA.

CA = (x+6)cm CA = (7 + 6)cm CA = 13cm Therefore, side CA is 13 cm

  1. Is CA ≅ EV?

EV = (4x-15)cm Given EV = (4(7) – 15)cm Substitute the value of x=7 and simplify EV = (21-15) cm Subtract EV = 13cm Therefore, side CA and EV are congruent

  1. How long is AV?

CE=(2y+1)cm , AV= (y+3)cm. CE ≅ AV In a parallelogram, opposite sides are congruent. 2y+1 = y+3 By substitution 2y+1-1 = y+3-1 Addition Property of Equality 2y-y = y-y+2 Addition Property of Equality

x+

4 x- 15

C A
E^ V
  1. Does angle D ≅ angle I? I = (x+10)^0 Given I = (15+10)^0 Substitute the value of x = 15 and simplify I = 25^0

Therefore, D and I are congruent.

  1. What is the measure of angle K? K and I are consecutive angles m K + m I = 180^0 In a parallelogram , two consecutive angles are supplementary. m K + 25^0 = 180^0 Substitute the measure of I m K + 25^0 - 25^0 = 180^0 - 25^0 Addition Property of Equality m K = 155^0

Therefore the measure of K = 155^0.

Property number 4 says that the diagonal in a parallelogram bisect each other_._

Let us try a look at this given example 3. Given a parallelogram FACE , diagonal FC and AE intersect at M.

EM = (4x-2)cm and AM = (x+7)cm F A

  1. How long is EM?
  2. Does EM ≅ AM?
  3. What is the length of AE?

To solve the problem used property number 4 which states that the diagonals of a

parallelogram bisect each other.

  1. How long is EM? Given : EM = (4x-2) cm and AM = (x+7) cm EM ≅ AM the diagonal in a parallelogram bisect each other 4x-2 = x+7 Substitute the measure of diagonal EM and AM 4x -2 + 2 = x + 7 +2 Addition Property of Equality 4x – x = x – x +9 Subtraction Property of Equality 3x = 9 3x = 9 3 3 x = 3

E C

M

By substitution EM = (4x-2) cm EM = (4(3) – 2) cm EM = (12 – 2) cm EM = 10 cm

Therefore, EM is 10 cm long

  1. Does EM ≅ AM? Given: EM = (4x-2) cm and AM = (x+7) cm EM = AM 4x - 2 = x + 7 4(3) - 2 = 3 + 7 12 – 2 = 10 10 = 10 Therefore, the length of EM and AM are congruent.
  2. What is the length of AE? AE = AM + AE AE = 10 cm + 10 cm AE = 20 cm the length of diagonal AE

Example 4. A rectangular lot has a width of (2b + 4) cm and (b + 7) cm and its length is (3a+ 5) cm and (2a + 16) cm. a. What is the width of the rectangular lot? b. What is the length of the rectangular lot? c. Find the perimeter of the lot.

Since a rectangle is a parallelogram then we can apply the property which states that in a parallelogram opposite sides are congruent.

Solution: a. What is the width of the rectangular lot? (2b + 4) cm = (b + 7) cm In a parallelogram, opposite side are congruent 2b + 4 = b+ 7 2b – b + 4 – 4 = b + 7 – 4 Addition Property of Equality b = 3 value of b By Substitution: (2b + 4) cm = (b + 7) cm (2(3) + 4) cm = (3 + 7) cm (6 + 4) cm = 10 cm 10 cm = 10 cm Therefore, the width of the rectangular lot is 10cm and its width are congruent.

Example 1. Quadrilateral HEAT is an isosceles trapezoid with HE ‖ TA. BD is its median. If HE=(x+10) cm , A = (3x – 2) cm and BD = 20 cm, find the length of the two bases.

Using property number 1 and 2:

  1. The Midsegment Theorem. The median of trapezoid is parallel(‖) to each base and its length is one half the sum of the lengths of the bases.
  2. The base angles of an isosceles trapezoid are congruent (≅). Solution: BD = 1(HE + TA) Formula 2 20cm = 1[(x+10)cm + (3x-2)cm] Substitute the value 2 2(20) = 1(2)(x+10 + 3x -2) Multiply both sides by 2 which is the 2 reciprocals of ½ to eliminate fraction. 40 = 2(4x + 8) Simplify 2 40 = 4x + 8 40 – 8 = 4x +8 - 8 Addition Property of Equality 32 = 4x 32 = 4x 4 4 x=8 the value of x

The two bases of the isosceles trapezoid.

  1. HE=(x+10)cm Given HE=(8+10)cm Substitute the value of x = 8 HE=18cm Length of base HE
  2. TA=(3x – 2)cm Given TA=(3(8) – 2)cm Substitute the value of x = 8 TA=(32 – 2)cm Simplify TA= 32cm Length of base TA Example 2. In this given example, we will apply the property number 3 and 4 of an isosceles trapezoid. Given: Quadrilateral LAKE is an isosceles trapezoid with pair of base angle: L and A, E and K.
  3. If the measure of E = (3x-17)^0 and K = (2x + 13)^0. a. What is the measure of E? b. Is the measure of E ≅ K? c. What is the measure of L?
  4. The measure of diagonal LK=(2x+3)cm and diagonal AE=(4x – 5)cm. What is the length of diagonal LK?Is the measure of LK≅AE?
E

L A

K

Solution:

  1. If the measure of E = (3x-17)^0 and K = (2x + 13)^0. a.What is the measure of E? E = K Base angles in an isosceles trapezoid ≅. (3x-17)^0 = (2x+13)^0 Substitute 3x-17 = 2x+ 3x-2x-17+17=2x-2x+13+17 Addition Property of Equality x = 30 Value of x

E = (3x-17)^0 Given E= (3(30)-17)^0 Substitute the value of x=30 and simplify E = (90-17)^0 E = 73^0 The measure of angle E

b. Is the measure of E ≅ K? K = (2x + 13)^0. Given K = (2(30) + 13)^0 Substitute the value of x= K = (60 + 13)^0 K = 73^0 The measure of angle E The measure of E ≅ K c. What is the measure of L? L is opposite to K L + K = 180^0 Opposite angles in an isoscele trapezoid are supplementary L + 73^0 = 180^0 Substitute L + 73^0 -73^0 = 180^0 - 73^0 Addition Property of Equality L = 180^0 - 73^0 Addition Property of Equality L = 107^0 The measure of L

  1. The measure of diagonal LK=(2x+3)cm and diagonal AE=(4x + 5)cm. What is the length of diagonal LK?Is the measure of LK≅AE? Solution: LK = AE Diagonals in an isosceles trapezoids congruent (4x+3)cm = (2x + 5)cm By Substitution 4x+3 = 2x + 5 4x-2x+3-3 = 2x-2x+5-3 Addition and Subtraction Property of Equality 2x= 2x= 2 2 x=1 value of x

To find the length of LK To find the length of AE LK = (4x+3)cm AE=(2x + 5)cm LK = (4(1)+3)cm AE=(2(1) + 5)cm LK = (4+3)cm AE=(2 + 5)cm LK = 7cm the length of LK AE= 7cm the length of AE Therefore diagonal LK and AE in an isosceles trapezoid are congruent.

Example 2. The area of the kite is 90cm^2 and the length of the diagonal is 18cm.How long is the other diagonal? Solution: A = 1 (d 1 d 2 ) Area of the kite is ½ the product of the 2 length of its diagonal. 90cm^2 = 1 (18cm) (d 2 ) By substitution 2 90cm^2 = 18cm (d 2 ) Simplify 2 90cm^2 = 9cm (d 2 ) 90cm^2 = 9cm (d 2 ) 9cm 9cm 10cm = d 2 d 2 = 10cm the length of the other diagonal in a kite.

What’s More

Activity 3: Try Me! Directions: Solve the following problems involving parallelograms, trapezoid and kite. State the property used to solve the problems. Write your answer on the space provided and use extra sheets for your solution.

  1. A quadrilateral MAID is a parallelogram.MA =(2x+3) cm and ID= (x + 6) cm, what is the length of the two sides?
  2. BEAR is a parallelogram with measure of angle B = (3x+1)^0 and the measure of angle A = (2x-5)^0. What is the measure of angle E?
  3. An isosceles trapezoid HIDE has a median XZ = 20cm.HI ‖DE, If HI = (8x-10) cm and DE = (15-x) cm, find the measure of the parallel sides.
  4. The diagonals of a kite are 8cm and 3cm. What is the area of the kite?
  5. A kite with an area of 70cm^2 and one diagonal measures 10cm and other diagonal is(2x+4) cm, find the length of the other diagonal?

What I Have Learned Activity 4. Paired Me Up Directions: Choose your answer in the box. Write the correct answer on the space provided.

480 56cm 800 7cm 45cm

____1. The diagonals of a kite have lengths of 14cm and 8cm. Find the area of the kite. ____2. If the measure of one angle of an isosceles trapezoid is x^0 and the angle opposite it is (x+20)^0 , what is the measure of the smaller angle? ____3. Liza wishes to bake a rectangular cake with the parallel side measures (3x+4) cm and (2x + 5) cm, what is the length of the side?

____4. An isosceles trapezoid MILD where MI ‖ DL with AB as median. If MI=(2x+7) cm, LD=(x-3) cm and AB= 34 cm. How long is DL?

____5. ABCD is a kite. AB is adjacent to BC and diagonal AC and BD intersect at E. If BEA =(15x)^0 and BAE = 8x-6, find the m ABC.

What I Can Do

Activity 5: Feel Me! Direction: Illustrate the problem and solve. Write your answer on the space provided and use extra sheets for your solution.

  1. Dexter wants to make rectangular swimming pool. If the two parallel sides has a corresponding measure of (4x-8) cm and (x+13) cm, what is the measure of the sides?
  2. Dr. James property is shaped like an isosceles trapezoid. Dr. James gave the contractor the following measurements, so that the contractor can build a wall around the entire property: base 1 = (2x – 5) km, base 2 = (x +7) km and a median of 7km and its leg measures 3km. What is the perimeter of wall?
  3. My father makes a kite with an area of 25m^2. One of its diagonals is 5m, what is the length of the other diagonal?

Assessment

Multiple Choice Test: Encircle the letter of the correct answer.

  1. Which property of a parallelogram is to be use to find the measure of its opposite angle? A. In a parallelogram, any two opposite sides are congruent. B. In a parallelogram, any two opposite angles are congruent. C. In a parallelogram, any two consecutive angles are supplementary D. The diagonals of a parallelogram bisect each other.
  2. LIFE is a kite. What are its two diagonals? A. LI and FE B. LE and FI C. IE and LF D. L and E
  3. In a parallelogram SITE, the length of SI=12cm.

What is the length of ET? A. 3cm B. 6cm C. 12cm D. 24cm

  1. If the measure of angle B in a parallelogram BITE is 120^0 , what is the measure of angle I? A. 10^0 B. 20^0 C. 40^0 D. 60^0
  2. A quadrilateral FERN is a parallelogram. If measure of FE= (3x + 3) cm and RN = (x +13) cm.

How long is RN? A. 5cm B. 8cm C. 10cm D. 18cm

Mathematics 9 3 RD^ QUARTER MODULE 4 Answer Key

What I know

  1. B
  2. A
  3. C
  4. D
  5. D
  6. D
  7. B
  8. A
  9. D
  10. D

What’s In Activity 1

  1. G
  2. D
  3. B
  4. C
  5. A
  6. F
  7. E

What’s New Activity 2

  1. T
  2. F-opposite
  3. F- supplementary
  4. F-half
  5. T
  6. T
  7. T
  8. T
  9. T
  10. T

What’s More Activity 3

  1. 9cm
  2. 130
  3. 30cm and 10cm
  4. 12cm^2
  5. 14cm

What I have learned Activity 4

  1. 56 cm
  2. 800
  3. 7cm
  4. 45cm
  5. 480

What I can do Activity 5

  1. 20cm
  2. 26km
  3. 10m

Assessment

  1. B
  2. C
  3. C
  4. D
  5. D
  6. C
  7. D
  8. D
  9. D
  10. C

References:

Bryant, Merden L., Leonides E. Bulalayao, Melvin M. Callanta, Jerry D. Cruz, et al. Mathematics Learner’s Material 9. Pasig City: Department of Education, 2014.

Bryant, Merden L., Leonides E. Bulalayao, Melvin M. Callanta, Jerry D. Cruz, et al.Mathematics Teachers Guide 9. Pasig City: Department of Education, 2014

Bernabe, Julieta G., Soledad Jose-Dilao and Fernando B Orines, Quadrilaterals Geometry, 1251 Gregorio Araneta Avenue, Quezon City: SD Publications Inc., 2009.

Development Team

Writer: ANALIZA S. LAPAD

Diplahan National High School

Editor/QA: Eugenio E. Balasabas

Ressme M. Bulay-og

Mary Jane I. Yeban

Reviewer: Gina I. Lihao

EPS-Mathematics

Illustrator: Layout Artist:

Management Team: Evelyn F. Importante

OIC-CID Chief EPS

Jerry c. Bokingkito

OIC-Assistant SDS

Aurelio A. Santisas, CESE

OIC- Assistant SDS

Jenelyn A. Aleman, CESO VI

OIC- Schools Division Superintendent