Prove the Area of Shaded Regions in Triangle with Semicircles Using Pythagorean Theorem

Math Problem Statement

In triangle $ABC,$ $\angle B = 90^\circ.$ Semicircles are constructed on sides $\overline{AB},$ $\overline{AC},$ and $\overline{BC},$ as shown below. Show that the total area of the shaded region is equal to the area of triangle $ABC.$

Asymptote code below [asy] unitsize(2 cm);

pair A = (1, 0); pair B = (7/25, 24/25); pair C = (-1, 0);

path AB = arc((16/25, 12/25), 3/5,-53.13, 180-53.13); path BC =arc((-9/25, 12/25), 4/5,36.87, 180+36.87); path AC = arc((0, 0), 1, 0, 180);

fill(AB--cycle, gray); fill(BC--cycle, gray); fill(AC--cycle, palegray); draw(arc((0, 0), 1, 0, 180)); draw(arc((16/25, 12/25), 3/5,-53.13, 180-53.13)); draw(arc((-9/25, 12/25), 4/5,36.87, 180+36.87)); draw(A--B--C--cycle); draw(rightanglemark(A, B, C, 3));

label("$A$", A, E); label("$B$", B, N); label("$C$", C, W); [/asy]

Solution

Given a right triangle $\triangle ABC$ with $\angle B = 90^\circ$ , semicircles are constructed on each of its sides. The goal is to prove that the total area of the shaded regions equals the area of the triangle $\triangle ABC$ .

Step 1: Define the sides of the triangle

Let:

$AB = c$ (the side opposite $\angle C$ ),
$BC = a$ (the side opposite $\angle A$ ),
$AC = b$ (the hypotenuse).

Step 2: Calculate the area of the triangle

The area of $\triangle ABC$ can be calculated using the formula for the area of a right triangle:

$\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times c \times a.$

Step 3: Find the area of each semicircle

Semicircle on $AB$ :
- Radius $r_1 = \frac{c}{2}$ .
- Area of the semicircle:
  
  $\text{Area}_{AB} = \frac{1}{2} \pi \left(\frac{c}{2}\right)^2 = \frac{1}{2} \pi \frac{c^2}{4} = \frac{\pi c^2}{8}.$
Semicircle on $BC$ :
- Radius $r_2 = \frac{a}{2}$ .
- Area of the semicircle:
  
  $\text{Area}_{BC} = \frac{1}{2} \pi \left(\frac{a}{2}\right)^2 = \frac{1}{2} \pi \frac{a^2}{4} = \frac{\pi a^2}{8}.$
Semicircle on $AC$ :
- Radius $r_3 = \frac{b}{2}$ .
- Area of the semicircle:
  
  $\text{Area}_{AC} = \frac{1}{2} \pi \left(\frac{b}{2}\right)^2 = \frac{1}{2} \pi \frac{b^2}{4} = \frac{\pi b^2}{8}.$

Step 4: Determine the shaded area

The shaded area is formed by subtracting the areas of the semicircles on sides $AB$ and $BC$ from the semicircle on the hypotenuse $AC$ :

$\text{Shaded Area} = \text{Area}_{AC} - (\text{Area}_{AB} + \text{Area}_{BC}).$

Substituting the values from above:

$\text{Shaded Area} = \frac{\pi b^2}{8} - \left(\frac{\pi c^2}{8} + \frac{\pi a^2}{8}\right) = \frac{\pi b^2}{8} - \frac{\pi (a^2 + c^2)}{8}.$

Step 5: Use the Pythagorean theorem

Since $\triangle ABC$ is a right triangle, by the Pythagorean theorem:

$b^2 = a^2 + c^2.$

Substituting $b^2$ in the shaded area formula:

$\text{Shaded Area} = \frac{\pi (a^2 + c^2)}{8} - \frac{\pi (a^2 + c^2)}{8} = \frac{\pi (a^2 + c^2) - \pi (a^2 + c^2)}{8} = \frac{a \cdot c}{2}.$

Thus the shaded area $ABC.$

Step 5: Relative question

How to find the shaded part of semicircle?
How to find perpendicular triangle?
How to find x part of coordinate triangle
How semicircle defines a new theorem?

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Math Problem Analysis

Mathematical Concepts

Geometry
Areas of Circles and Triangles
Pythagorean Theorem

Formulas

Area of a right triangle: Area = 1/2 * base * height
Area of a semicircle: Area = 1/2 * π * r^2
Pythagorean Theorem: b^2 = a^2 + c^2

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 9-12

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