Examples with solutions for Area of a Triangle: Using ratios for calculation

Exercise #1

The area of trapezoid ABCD is X cm².

The line AE creates triangle AED and parallelogram ABCE.

The ratio between the area of triangle AED and the area of parallelogram ABCE is 1:3.

Calculate the ratio between sides DE and EC.

AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

To calculate the ratio between the sides we will use the existing figure:

AAEDAABCE=13 \frac{A_{AED}}{A_{ABCE}}=\frac{1}{3}

We calculate the ratio between the sides according to the formula to find the area and then replace the data.

We know that the area of triangle ADE is equal to:

AADE=h×DE2 A_{ADE}=\frac{h\times DE}{2}

We know that the area of the parallelogram is equal to:

AABCD=h×EC A_{ABCD}=h\times EC

We replace the data in the formula given by the ratio between the areas:

12h×DEh×EC=13 \frac{\frac{1}{2}h\times DE}{h\times EC}=\frac{1}{3}

We solve by cross multiplying and obtain the formula:

h×EC=3(12h×DE) h\times EC=3(\frac{1}{2}h\times DE)

We open the parentheses accordingly:

h×EC=1.5h×DE h\times EC=1.5h\times DE

We divide both sides by h:

EC=1.5h×DEh EC=\frac{1.5h\times DE}{h}

We simplify to h:

EC=1.5DE EC=1.5DE

Therefore, the ratio between is: ECDE=11.5 \frac{EC}{DE}=\frac{1}{1.5}

Answer

1:1.5 1:1.5

Exercise #2

Since the side BC is 13 \frac{1}{3} side AE.

Calculate the area of the triangle:

666AAABBBCCCEEE

Video Solution

Answer

6

Exercise #3

Since the side BC is 14 \frac{1}{4} side AE.

Calculate the area of the triangle:

121212AAABBBCCCEEE

Video Solution

Answer

18

Exercise #4

Since the side BC is 15 \frac{1}{5} side AE.

Calculate the area of the triangle:

101010AAABBBCCCEEE

Video Solution

Answer

10

Exercise #5

Since the side BC is 23 \frac{2}{3} side AE.

Calculate the area of the triangle:

999AAABBBCCCEEE

Video Solution

Answer

27

Exercise #6

Since the side BC is 58 \frac{5}{8} side AE.

Calculate the area of the triangle:

161616AAABBBCCCEEE

Video Solution

Answer

80

Exercise #7

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 #8

The area of trapezoid ABCD

is 30 cm².

The line AE creates triangle AED and parallelogram ABCE.

The ratio between the area of triangle AED and the area of parallelogram ABCE is 1:2.

AAABBBCCCDDDEEE

Calculate the ratio between sides DE and EC.

Video Solution

Answer

1