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
$\frac{6}{9} = \frac{y}{12} \\ y = 8$
(6+3)*12/2 - 6*8 /2 = 30

Answer 2

Answer:

30 units ^2

Step-by-step 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