Question

pythagorean theorem
consider any right triangle where the short sides have lengths $a$ and $b$ and the hypotenuse has length $c$.
then the pythagorean theorem is $a^{2}+b^{2}=c^{2}$

example
using the theorem, you have $3^{2}+4^{2}=c^{2}$
$9 + 16 = c^{2}$
$25 = c^{2}$
$sqrt{25}=c$
$5 = c$

example
using the theorem, you have $5^{2}+b^{2}=13^{2}$
$25 + b^{2}=169$
$b^{2}=169 - 25$
$b^{2}=144$
$b=sqrt{144}$
$b = 12$

using the pythagorean theorem, calculate the value of the missing length. draw an arrow to the matching answer in the middle.

1. right triangle with legs 9cm, 12cm, missing hypotenuse $c$
2. right triangle with legs 15cm, 20cm, missing hypotenuse $c$
3. right triangle with one leg 30cm, hypotenuse 78cm, missing leg $b$
4. right triangle with one leg 45cm, hypotenuse 51cm, missing leg $a$
5. right triangle with one leg 32cm, hypotenuse $c$, other leg 60cm
6. right triangle with hypotenuse 34cm, one leg 30cm, missing leg $a$
7. right triangle with one leg 27cm, hypotenuse 36cm, missing leg $c$
8. right triangle with legs 25cm, 60cm, missing hypotenuse $c$
9. right triangle with one leg 20cm, hypotenuse $c$, other leg 48cm
10. right triangle with hypotenuse 75cm, one leg 65cm, missing leg $a$

answer options:
16cm, 65cm, 52cm, 15cm, 25cm, 68cm, 45cm, 24cm, 72cm, 40cm

Explanation:

Step1: Calculate missing hypotenuse (9,12)

$c=\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$

Step2: Calculate missing hypotenuse (15,20)

$c=\sqrt{15^2+20^2}=\sqrt{225+400}=\sqrt{625}=25$

Step3: Calculate missing leg (30,78)

$b=\sqrt{78^2-30^2}=\sqrt{6084-900}=\sqrt{5184}=72$

Step4: Calculate missing leg (45,51)

$a=\sqrt{51^2-45^2}=\sqrt{2601-2025}=\sqrt{576}=24$

Step5: Calculate missing hypotenuse (32,60)

$c=\sqrt{32^2+60^2}=\sqrt{1024+3600}=\sqrt{4624}=68$

Step6: Calculate missing leg (30,34)

$a=\sqrt{34^2-30^2}=\sqrt{1156-900}=\sqrt{256}=16$

Step7: Calculate missing hypotenuse (27,36)

$c=\sqrt{27^2+36^2}=\sqrt{729+1296}=\sqrt{2025}=45$

Step8: Calculate missing hypotenuse (25,60)

$c=\sqrt{25^2+60^2}=\sqrt{625+3600}=\sqrt{4225}=65$

Step9: Calculate missing hypotenuse (20,48)

$c=\sqrt{20^2+48^2}=\sqrt{400+2304}=\sqrt{2704}=52$

Step10: Calculate missing leg (75,85)

$a=\sqrt{85^2-75^2}=\sqrt{7225-5625}=\sqrt{1600}=40$

Answer:

1. Triangle (9cm, 12cm) → 15cm
2. Triangle (15cm, 20cm) → 25cm
3. Triangle (30cm, 78cm) → 72cm
4. Triangle (45cm, 51cm) → 24cm
5. Triangle (32cm, 60cm) → 68cm
6. Triangle (30cm, 34cm) → 16cm
7. Triangle (27cm, 36cm) → 45cm
8. Triangle (25cm, 60cm) → 65cm
9. Triangle (20cm, 48cm) → 52cm
10. Triangle (75cm, 85cm) → 40cm