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Problem Breakdown and Solutions:

1. Calculate the hypotenuse of a right triangle (Question 1):

   • Use the Pythagorean theorem: $c = \sqrt{a^2 + b^2}$.
   • For each triangle:
     • (a) $a = 7 \text{ cm}, b = 24 \text{ cm}$ $c = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \text{ cm}$
     • (b) $a = 3.2 \text{ cm}, b = 2.4 \text{ cm}$ $c = \sqrt{3.2^2 + 2.4^2} = \sqrt{10.24 + 5.76} = \sqrt{16} = 4 \text{ cm}$

2. Calculate the unknown leg of a right triangle (Question 2):

   • Again, use the Pythagorean theorem, but solve for the missing leg: $b = \sqrt{c^2 - a^2}$
   • For each triangle:
     • (a) $c = 20 \text{ cm}, a = 12 \text{ cm}$ $b = \sqrt{20^2 - 12^2} = \sqrt{400 - 144} = \sqrt{256} = 16 \text{ cm}$
     • (b) $c = 8 \text{ cm}, a = 4.5 \text{ cm}$ $b = \sqrt{8^2 - 4.5^2} = \sqrt{64 - 20.25} = \sqrt{43.75} \approx 6.61 \text{ cm}$

3. Determine if a triangle is a right triangle based on side lengths (Question 3):

   • Check if the triangle satisfies $a^2 + b^2 = c^2$.
     • (a) $14 \text{ cm}, 48 \text{ cm}, 50 \text{ cm}$: $14^2 + 48^2 = 196 + 2304 = 2500 \quad \text{and} \quad 50^2 = 2500$ This is a right triangle.
     • (b) $32 \text{ cm}, 25.5 \text{ cm}, 25.7 \text{ cm}$: $32^2 + 25.5^2 = 1024 + 650.25 = 1674.25 \quad \text{but} \quad 25.7^2 = 660.49$ This is not a right triangle.

4. Calculate the length of the perpendicular leg in a right triangle based on a diagonal angle (Question 4):

   • Given $AC = 30 \text{ cm}$ and $\angle ABC = 60^\circ$, find the leg opposite and adjacent to the angle.
   • Since it forms a $30^\circ$-$60^\circ$-$90^\circ$ triangle, the ratio of sides is $1 : \sqrt{3} : 2$.
     • The length of $BC = \frac{AC}{2} = \frac{30}{2} = 15 \text{ cm}$.
     • The length of the other leg, $AB = 15\sqrt{3} \approx 25.98 \text{ cm}$.

5. Find the angles in isosceles triangles using the given heights or angles (Question 5):

   • For each part, use the properties of isosceles triangles and supplementary angle properties.
   • (a) $\angle ADC = \angle ABC = 120^\circ$.
   • (b) Use the given information to deduce remaining angles based on triangle properties.

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Follow-up Questions:

1. How would you solve for an unknown angle in a triangle if only two sides are given?
2. How do you calculate the area of a right triangle given its two legs?
3. Can you find the missing side of a triangle if only one side and one angle are known?
4. How do the side ratios work in 30-60-90 triangles, and why do they hold?
5. Can you use trigonometric functions (sin, cos, tan) to solve these triangles?

Tip:

Remember, in any right triangle, the Pythagorean theorem can help confirm side lengths, and trigonometric functions can help determine angles and other unknown sides if one side and one angle are known.