Question

REALLY IMPORTANT!!! PLEASE HELP!!! If triangle ABC is a right triangle and BC is a semicircle, find the total area of the figure below. what is the diameter of the semi-circle? _____ units. Area=____ units squared (rounded to the nearest hundredth. ) side AB is 3 units long. Side AC is 4 units long.

Answer 1

Answer:

The diameter of the semi-circle is 5 units

A = 9.82 units²

Step-by-step explanation:

In a circle the inscribed angle subtended by an arc of measure 180° (semi-circle) is a right angle

That means the angle whose vertex lies on a semi circle is a right angle

∵ Arc BC is a semi-circle

∵ Δ ABC is a right triangle

- That means BC is the diameter of the semi-circle and angle A

   is the right angle because A lies on the arc BC

∵ In Δ ABC

∵ m∠A = 90°

∵ AB = 3 units

∵ AC = 4 units

- Use Pythagoras theorem to find the length of BC

∵ (BC)² = (AB)² + (AC)²

∴ (BC)² = (3)² + (4)²

∴ (BC)² = 9 + 16

∴ (BC)² = 25

- Take √  for both sides

∴ BC = 5 units

∵ BC is the diameter of the semi-circle

∴ The diameter of the semi-circle is 5 units

∵ Area of the circle = πr²

- Multiply it by half to find the area of the semi-circle

∴ Area of semi-circle = $\frac{1}{2}$ πr²

∵ The length of the radius is half the length of the diameter

   of the circle

∴ r = $\frac{1}{2}$ (5) = 2.5 units

∴ A = $\frac{1}{2}$ π (2.5)²

∴ A = 9.817477042

- Round it to the nearest hundredth

∴ A = 9.82 units²