Question

using the theorem, you have $5^{2}+b^{2}=13^{2}$
$25 + b^{2}=169$
$b^{2}=169 - 25$
$b^{2}=144$
$b = \sqrt{144}$
$b = 12$
arrows to the matching answer in the middle.

1. a right triangle with hypotenuse 34cm, one right side 30cm, find the length of side $a$
2. a right triangle with one right side 36cm, the other right side 27cm, find the length of side $c$
3. a right triangle with one right side 25cm, the other right side 60cm, find the length of side $c$
4. a right triangle with one right side 20cm, the other right side 48cm, find the length of side $c$
5. a right triangle with hypotenuse 85cm, one right side 75cm, find the length of side $a$

Explanation:

Step1: Apply Pythagoras theorem for $\boldsymbol{a}$

For right triangles, $a^2 + b^2 = c^2$. Rearranged: $a = \sqrt{c^2 - b^2}$
First triangle: $a = \sqrt{34^2 - 30^2} = \sqrt{1156 - 900} = \sqrt{256} = 16$

Step2: Apply Pythagoras theorem for $\boldsymbol{c}$

Formula: $c = \sqrt{a^2 + b^2}$
Second triangle: $c = \sqrt{36^2 + 27^2} = \sqrt{1296 + 729} = \sqrt{2025} = 45$

Step3: Calculate third triangle's $\boldsymbol{c}$

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

Step4: Calculate fourth triangle's $\boldsymbol{c}$

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

Step5: Calculate fifth triangle's $\boldsymbol{a}$

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

Answer:

1. $a = 16$ cm
2. $c = 45$ cm
3. $c = 65$ cm
4. $c = 52$ cm
5. $a = 40$ cm