Area of a Triangle: Using additional geometric shapes

Question 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.

Question 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?

Question 3

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

its width $$ x+2y $$

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.

Question 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.

Question 5

ABCD is a parallelogram.

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

Express the area of triangle EBC in terms of X.

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.

Video Solution

Step-by-Step Solution

Let's calculate side BG using the Pythagorean theorem:

$BG^2+GC^2=BC^2$

We'll substitute the known data:

$BG^2+3^2=5^2$

$BG^2+9=25$

$BG^2=16$

$BG=\sqrt{16}=4$

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

$5\times4=20$

Answer

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?

Video Solution

Step-by-Step Solution

Let's calculate the area of triangle ABC:

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

Multiply by 2:

$16x=(x+4)x$

Divide by x:

$16=x+4$

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

$16-4=x$

$12=x$

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

$16\times12=192$

Answer

192

Exercise #3

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

its width $x+2y$

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

Step-by-Step Solution

Let's draw a line in the middle of the drawing that divides the house into 2

Meaning it divides the triangle and the rectangular part.

The 2 lines represent the heights in both shapes.

If we connect the height of the roof with the height of the rectangular part, we get the total height

Let's put the known data in the formula:

$\frac{1}{2}h_{\text{square}}+h_{square}=12x+9$

$\frac{3}{2}h_{\text{square}}=12x+9$

We'll multiply by two thirds and get:

$h_{\text{square}}=\frac{2(12x+9)}{3}=2(4x+3)$

$h_{\text{square}}=8x+6$

If the height of the triangle equals half the height of the rectangular part, we can calculate it using the following formula:

$h_{\text{triangle}}=\frac{1}{2}(8x+6)=4x+3$

Now we can calculate the area of the triangular part:

$\frac{(x+2y)\times(4x+3)}{2}=\frac{4x^2+3x+8xy+6y}{2}=2x^2+1.5x+4xy+3y$

Now we can calculate the rectangular part:

$(x+2y)\times(8x+6)=8x^2+6x+16xy+12y$

Now let's combine the triangular area with the rectangular area to express the total area of the shape:

$S=2x^2+1.5x+4xy+3y+8x^2+6x+16xy+12y$

$S=10x^2+20xy+7.5x+15y$

Answer

$3x^2+8xy+1\frac{1}{2}x+4y^2+3y$

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.

Video Solution

Answer

32 cm²

Exercise #5

ABCD is a parallelogram.

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

Express the area of triangle EBC in terms of X.

Video Solution

Answer

$3.5X + 7 cm²$

Question 1

ACDE is a parallelogram.

DE = 12

A semicircle is placed on side FB.

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

Question 2

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 $\pi=3.14$

Question 3

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.

Question 4

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

Question 5

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?

Exercise #6

ACDE is a parallelogram.

DE = 12

A semicircle is placed on side FB.

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

Video Solution

Answer

$36\sqrt{2}$ cm²

Exercise #7

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 $\pi=3.14$

Video Solution

Answer

17.1 cm².

Exercise #8

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.

Video Solution

Answer

$5\sqrt{14}+30$ cm².

Exercise #9

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

Video Solution

Answer

8 cm²

Exercise #10

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?

Video Solution

Answer

$36-\text{4}.5\pi$ cm².

Question 1

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

AB=7 DC=13

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

How large is the area of the triangle AFD

Exercise #11

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

AB=7 DC=13

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

How large is the area of the triangle AFD

Video Solution

Answer

25 cm²..