Question

In the figure, what is the area of the shaded region?

Answer 1

Use Pythagoras theorem

`{(6 + 3)}^(2) + {x}^(2) = {15}^(2) \\ {x}^(2) = 144 \\ x = 12`
Two triangles are similar

`(6)/(9) = (y)/(12) \\ y = 8`
(6+3)*12/2 - 6*8 /2 = 30

Answer 2

30 units ^2

Explanation:

To find the area of the shaded region, we find find the area of the large triangle and subtract the area of the unshaded triangle.

A of large triangle = 1/2 b*h

height = (6+3) = 9

The base is found by using the pythagorean theorem c^2 = a^2 + b^2

We need to find b^2

c^2 -a^2 = b^2

taking the square root on each side

sqrt(c^2 -a^2) = sqrt(b^2)

the base = sqrt(c^2 -a^2)

= sqrt( 15^2 - 9^2)

= sqrt(225-81)

= sqrt(144)

=12

Now that we know the base and the height, we can find the area

A of large triangle = 1/2 b*h

= 1/2 * 12 * 9

= 6*9 = 54

Using the rule of similar triangles

9 6

---- = ----------

12 base

We can use cross products to find the base of the smaller triangle

9* base = 12*6

9* base = 72

Divide by 9 on each side

base = 72/9 = 8

Now we can find the area of the smaller triangle

base = 8 and height = 6

A of smaller triangle = 1/2 b*h

= 1/2 *8 * 6

= 4*6 = 24

Area of the shaded region = Area large triangle - Area of small triangle

= 54-24

= 30