CHAPTER 13: PYTHAGORAS' THEOREM

CHAPTER 13: PYTHAGORAS' THEOREM

BAB 13: TEOREM PYTHAGORAS

HYPOTENUSE

HIPOTENUS

• Side that is opposite to the right angle.

• Sisi yang bertentangan dengan sudut tegak

• The longest side in the right-angled triangle

• Sisi yang terpanjang dalam segi tiga bersudut tegak itu.

Side

Sisi

• AC @ CA = b

• AB @ BA = c

• BC @ CB = a

• The longest side in triangle ABC is AC. Hence, AC is the hypotenuse.

• Sisi terpanjang dalam segi tiga ABC ialah AC. Maka, AC ialah hypotenus

• Capital letters are referring to the label or name of the side. Example: AC

• Huruf besar merujuk kepada label atau nama sisi. Contoh: AC

• Small letters are referring to the length of the side. Example: b = 5cm

• Huruf kecil merujuk kepada panjang sisi. Contoh: b = 5cm

• AC = b = 5cm

Multiple Choice

Based on the diagram given, identify the hypotenuse.

Berdasarkan gambar yang diberi, kenal pasti hipotenus.

• RP

• QR

• PQ

• QP

Multiple Choice

Based on the diagram give, identify the hypotenuse.

Berdasarkan gambar rajah yang diberi, kenal pasti hipotenus.

• NP

• MN

• MP

• PN

Multiple Choice

Based on the diagram give, identify the hypotenuse.

Berdasarkan gambar rajah yang diberi, kenal pasti hipotenus.

• JK

• LK

• KJ

• LJ

Multiple Choice

Based on the diagram give, identify the hypotenuse.

Berdasarkan gambar rajah yang diberi, kenal pasti hipotenus.

• d

• e

• f

Multiple Choice

Based on the diagram give, identify the hypotenuse.

Berdasarkan gambar rajah yang diberi, kenal pasti hipotenus.

• s

• t

• u

Relationship between the sides of a right-angled triangle

Hubungan antara sisi segi tiga bersudut tegak

•  $Area\ C=Area\ B+Area\ A$  

•  $Luas\ C=Luas\ B+Luas\ C$  

•  $c\times c=\left(b\times b\right)+\left(c\times c\right)$  

•  $c^2=b^2+a^2$  

Pythagoras Teorem

Teorem Pitagoras

• Must Remember the formula

•  $AC^2=AB^2+BC2$  

•  $b^2=a^2+c^2$  

•  $c^2=b^2-a^2$  

•  $a^2=b^2-c^2$  

• *  $hypotenuse^2=right\ \ ^2+left^2$  

•  $hipotenus^2=kanan^2+kiri^2$  

Multiple Choice

State the relationship between the length of the following right-angled triangle

• $NP^2=MN^2+MP^2$

• $MN^2=NP^2+MP^2$

• $PM^2=MN^2-NP^2$

• $MP^2=MN^2+NP^2$

Multiple Choice

State the relationship between the length of the following right-angled triangle

•  $PQ^2=PR^2+QR^2$ 

•  $PR^2=PQ^2+QR^2$ 

•  $QR^2=PQ^2-PR^2$ 

•  $QR^2=PQ^2+PR^2$ 

Multiple Choice

State the relationship between the length of the following right-angled triangle

•  $PQ^2=PR^2-QR^2$ 

•  $PR^2=PQ^2-QR^2$ 

•  $QR^2=PQ^2-PR^2$ 

•  $QR^2=PQ^2+PR^2$ 

Multiple Choice

State the relationship between the length of the following right-angled triangle

•  $t^2=s^2-u^2$ 

•  $s^2=u^2-t^2$ 

•  $t^2=s^2+u^2$ 

•  $s^2=t^2+u^2$ 

Multiple Choice

State the relationship between the length of the following right-angled triangle

•  $e^2=f^2-d^2$ 

•  $f^2=d^2+e^2$ 

•  $d^2=f^2-e^2$ 

•  $e^2=d^2-f^2$ 

Determining the length of the unknown side of a right-angled triangle

Menentukan panjang sisi yang tidak diketahui bagi suatu segi tiga bersudut tegak

Tips!

1) Identify the hypotenuse.

1) Tentukan hipotenus.

2) Apply pythagoras' theorem (formula)

2) Aplikasikan teorem pythagoras (formula)

Example 1:

Contoh 1:

Calculate the value of x. / Hitung nilai x.
Solution. / Penyelesaian.

 $hypotenuse^2=a^2+c^2$  
 $17^2=x^2+15^2$  
 $x^2=17^2-15^2$  
 $x^2=289-225$  
 $x^2=64$  
 $x=\sqrt{64}$  
 $x=8$  

Example 2:

Contoh 2:

Calculate the value of x./ Hitung nilai x.
Solution./ Penyelesaian.

 $x^2=12^2+5^2$  
 $x^2=144+25$  
 $x^2=169$  
 $x=\sqrt{169}$  
 $x=13$  

Example 3:

Contoh 3:

Calculate the value x./ Hitung nilai x.
Solution./ Penyelesaian.

 $5^2=4^2+x^2$  
 $x^2=5^2-4^2$  
 $x^2=25-16$  
 $x^2=9$  
 $x=\sqrt{9}$  
 $x=3$  

Multiple Choice

Find the value of x.

Hitung nilai x.

• 5

• 6

• 7

• 8

Multiple Choice

Find the value of x.

Cari nilai x.

• 11

• 12

• 13

• 14

Multiple Choice

Find the value of x.

Cari nilai x.

• 28

• 36

• 45

• 55

Example 4.

Contoh 4.

Calculate the length of PQ in the following diagram. /Hitung panjang PQ dalam rajah yang berikut.

Solution / Penyelesaian
Step 1: Find the length of PS 
Langkah 1: Cari panjang PS
 $PS^2=8^2+6^2$  
 $PS^2=64+36$  
 $PS^2=100$  
 $PS=\sqrt{100}$  
 $PS=10$  

Step 2: Find PR. Since we have found the value of PS=10 and SR=5, just add both values to get the length of PR / Langkah 2: Cari PR. Setelah kita mendapat nilai PS=10 dan diberi SR=5, maka kita hanya perlu menambah kedua-dua nilai.

 $PR=PS+SR$  
 $PR=10+5$  
 $PR=15$  

Step 3: Now, we can imagine the diagram like this to find the length of PQ./ Langkah 3: Sekarang, anda boleh membayangkan rajah seperti ini untuk memudahkan mencari panjang PQ.

 $PQ^2=PR^2+RQ^2$  
 $PQ^2=15^2+8^2$  
 $PQ^2=225+64$  
 $PQ^2=289$  
 $PQ=\sqrt{289}$  
 $PQ=17$  

Example 5

Contoh 5

Calculate the length of PQ./ Hitung panjang PQ.
Solution. / Penyelesaian.
* Notes: '_' at PQ and QR mean both have the same length.
Nota: '_' pada PQ dan QR bermaksud kedua-duanya mempunyai panjang yang sama.

 $PQ=QR$  

Step 1: Find PR./Langkah 1: Cari PR

 $SR^2=SP^2+PR^2$  
 $30^2=24^2+PR^2$  
 $30^2-24^2=PR^2$  
 $900-576=PR^2$  
 $324=PR^2$  
 $\sqrt{324}=PR$  
 $18=PR$  
 $PR=18$  

Step 2:  $PR=PQ+QR$ ,  $PR=18$  , and  $PQ=QR$  (same length)

Langkah 2:  $PR=PQ+QR$  ,  $PR=18$  , dan  $PQ=QR$  (sama panjang)

 $PR=PQ+PQ$  
 $PR=2PQ$  
 $PQ=\frac{PR}{2}$  
 $PQ=\frac{18}{2}$  
 $PQ=9$  

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End of 13.1

Next topic 13.2