The values of the sides for the smaller triangle are: x =4 cm² and y=5 cm².
Similar Triangles
Two triangles are classified as similar when their angles are congruent and their sides are proportional.
Ratios
Ratios can represent a relation between two sides of a polygon. When the ratio is greater than 1, it means that the ratio is written as a fraction between the side of the greater polygon and the side of the smaller polygon. Otherwise, if the ratio is less than 1, it means that the ratio is written as a fraction between the side of the smaller polygon and the side of the greater polygon.
The figure shows that the triangles are similar, so you can write a ratio from their sides.
The question gives:
- ratio of sides = 4:5 since [tex]\frac{smaller\; side}{greater\; side} = \frac{8\div 2 }{10\div 2}=\frac{4}{5}[/tex]
- area of the smaller triangle= 10 cm² . Then, the area of the triangle is [tex]\frac{xy}{2} =10[/tex]
From the ratio of sides, you have:
[tex]\frac{smaller\; side}{greater\; side} = \frac{8\div 2 }{10\div 2}=\frac{4}{5}=\frac{x}{y}[/tex]
Thus,
[tex]4y=5x\\ \\ y=\frac{5x}{4}[/tex]
Replace [tex]y=\frac{5x}{4}[/tex] in the equation for area of the smaller triangle [tex]\frac{xy}{2} =10[/tex], then you have:
[tex]\frac{x*\frac{5x}{4} }{2} =10\\ \\ \\ \frac{5x^2}{4} =20\\\\ 5x^2=80\\ \\ x^2=16\\ \\ x=\sqrt{16} \\ \\ x=4[/tex]
If x=4 and the area of the smaller triangle is equal to 10 cm², you have:
[tex]\frac{xy}{2} =10\\ \\ \frac{4y}{2} =10\\ \\ 4y=20\\ y=5[/tex]
Therefore, x=4 cm and y= 5 cm. Note that from these values for x and y, you find the given area (10 cm²). See below:
[tex]\frac{xy}{2} =10\\ \\ \frac{4*5}{2} =10\\ \\ \frac{20}{2} =10\\ \\ 10=10[/tex]
Read more about ratio of sides here:
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