Examples with solutions for Area of a Triangle: Using additional geometric shapes

Exercise #1

The trapezoid ABCD and the rectangle ABGE are shown in the figure below.

Given in cm:

AB = 5

BC = 5

GC = 3

Calculate the area of the rectangle ABGE.

555555333AAABBBCCCDDDEEEGGG

Video Solution

Step-by-Step Solution

Let's calculate side BG using the Pythagorean theorem:

BG2+GC2=BC2 BG^2+GC^2=BC^2

We'll substitute the known data:

BG2+32=52 BG^2+3^2=5^2

BG2+9=25 BG^2+9=25

BG2=16 BG^2=16

BG=16=4 BG=\sqrt{16}=4

Now we can calculate the area of rectangle ABGE since we have the length and width:

5×4=20 5\times4=20

Answer

20

Exercise #2

Given the rectangle ABCD

Given BC=X and the side AB is larger by 4 cm than the side BC.

The area of the triangle ABC is 8X cm².

What is the area of the rectangle?

S=8XS=8XS=8XX+4X+4X+4XXXAAABBBCCCDDD

Video Solution

Step-by-Step Solution

Let's calculate the area of triangle ABC:

8x=(x+4)x2 8x=\frac{(x+4)x}{2}

Multiply by 2:

16x=(x+4)x 16x=(x+4)x

Divide by x:

16=x+4 16=x+4

Let's move 4 to the left side and change the sign accordingly:

164=x 16-4=x

12=x 12=x

Now let's calculate the area of the rectangle, multiply the length and width where BC equals 12 and AB equals 16:

16×12=192 16\times12=192

Answer

192

Exercise #3

The area of the triangle below is equal to 10 cm² and its height is 5 times greater than its base.

Calculate X.

101010xxx

Video Solution

Answer

x=2 x=2

Exercise #4

The parallelogram ABCD and the triangle BCE are shown below.

CE = 7
DE = 15

The area of the triangle BCE is equal to 14 cm².

Calculate the area of the parallelogram ABCD.

S=14S=14S=14777BBBCCCEEEAAADDD15

Video Solution

Answer

32 cm²

Exercise #5

AD is perpendicular to BC

AD=3

The area of the triangle ABC is equal to 7 cm².

BC is the diameter of the circle on the drawing

What is the area of the circle?
Replace π=3.14 \pi=3.14

S=7S=7S=7333AAABBBCCCDDD

Video Solution

Answer

17.1 cm².

Exercise #6

ABCD is a parallelogram.

AD is the diameter of a circle that has a circumference of 7π 7\pi cm.

Express the area of triangle EBC in terms of X.

X+2X+2X+2EEECCCBBBDDDAAA

Video Solution

Answer

3.5X+7cm2 3.5X + 7 cm²

Exercise #7

ACDE is a parallelogram.

DE = 12

A semicircle is placed on side FB.

If the area of the semicircle is 9π 9\pi , then what is the area of triangle ABC?

AAABBBCCCFFFEEEDDD

Video Solution

Answer

362 36\sqrt{2} cm²

Exercise #8

Given the triangle ABC and the deltoid ADEF

The height of the triangle is 4 cm

The base of the triangle is greater by 2 than the height of the triangle.

Segment FD cut to the middle

Calculate the area of the deltoid ADEF

444AAABBBCCCFFFEEEDDD

Video Solution

Answer

8 cm²

Exercise #9

Given the triangle ABC when the base BC a semi-circle is drawn

The radius of the circle is equal to 3 cm and its center is the point D

Given AE=3 ED

What is the area of the dotted shape?

333BBBDDDCCCAAAEEE

Video Solution

Answer

364.5π 36-\text{4}.5\pi cm².

Exercise #10

The height of the house in the drawing is 12x+9 12x+9

its width x+2y x+2y

Given the ceiling height is half the height of the square section.

Express the area of the house shape in the drawing band x and and.

Video Solution

Answer

3x2+8xy+112x+4y2+3y 3x^2+8xy+1\frac{1}{2}x+4y^2+3y

Exercise #11

Look at the triangle in the figure.

AD is used to form a semicircle with a radius of 2.5 cm.

Calculate the area of the triangle ABC.

999AAACCCBBBDDD12

Video Solution

Answer

514+30 5\sqrt{14}+30 cm².

Exercise #12

Given the trapezoid ABCD whose area is equal to 50 cm².

AB=7 DC=13

The area of the circle whose diameter FC is 2.25π 2.25\pi

How large is the area of the triangle AFD

S=50S=50S=50777131313AAABBBCCCDDDFFFEEE

Video Solution

Answer

25 cm²..