Right Triangles

Classify the following angles labelled I to III.

1. Right triangles
2. Non-right triangles
3. Isosceles right triangles
4. Scalene right triangles

Solution:

We can see that figure I is a right triangle because it has one of its angles equal to 90°. However, the indications on its sides show that no two of its sides are equal. This means that figure I is a scalene right triangle.

However, in figure II, none of its angles equals 90º. Hence figure II is a non-right triangle.

Likewise what we have in figure I, figure III has one of its angles equal to 90°. This makes it a right triangle. Unlike figure I, figure III has a 45º angle, which means that the third angle would also be 45°. Therefore, this implies that figure III is an isosceles right triangle since it does not just possess one of its angles equal to 90° but the two other angles are equal. Hence the right response to this is question is,

a. Right triangles - I and III

b. Non-right triangle - II

c. Isosceles right triangle - III

d. Scalene right triangle - I

Find the perimeter of the triangle.

Solution:

The perimeter of the triangle is equal to the sum of the lengths of its sides. Thus,

$P=3+4+5=12cm$

A right triangle cement block with sides 5 cm, 13 cm, and 12 cm is used to cover up a square lawn with a side length of 30 cm. How many right triangles are needed to cover the lawn?

Solution:

We need to determine the surface area of the square lawn. We let l be the side length of the square lawn so l = 30m,

${Area}_{squarelawn}=l^2={30}^2=900m^2$

In order to know the number of right triangles that would cover up the square lawn, we should calculate the area of each right triangle that would occupy in order to fill the square.

${Area}_{righttriangle}=\frac{1}{2}\times base\times height=\frac{1}{2}\times 12\times 5=30c{m}^2$

Now the area of the right triangle and the square has been calculated, we can now determine how many of the right-triangular cement blocks can be found on the square lawn.

$Numberofcementblock=\frac{Areaofsquarelawn}{Areaofrightangledcementblock}=\frac{Are{a}_{squarelawn}}{Are{a}_{righttriangle}}$

But first, we need to convert m² to cm² by recalling that

$$\begin{gathered} 100cm=1m \\ {(100cm)}^2={(1m)}^2 \\ 10000c{m}^2=1m^2 \\ 900m^2=9000000c{m}^2 \end{gathered}$$

Thus,

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{cement}\mathrm{block}=\frac{9000000c{m}^2}{30c{m}^2} \\ \mathrm{Number}\mathrm{of}\mathrm{cement}\mathrm{block}=300000 \end{gathered}$$

Therefore, one would need 300,000 right triangles (5 cm by 12 cm by 13 cm) to cover up a 30 m length square lawn.

The figure below comprises two right triangles which are joined together. If the hypotenuse of the bigger right triangle is 15 cm, find the ratio of the area of the bigger to smaller right triangle.

Solution:

Since the length of the hypotenuse of the bigger right triangle is 15 cm, the hypotenuse of the smaller right triangle is

$20cm-15cm=5cm$

We need to find the area of the bigger right triangle, which is A_b, and calculated it as:

$$\begin{gathered} Area=\frac{1}{2}\times base\times height \\ {A}_b=\frac{1}{2}\times 9cm\times 12cm \\ {A}_b=\frac{1}{\cancel{2}}\times 9cm\times {}^6\cancel{12}cm \\ {A}_b=9cm\times 6cm \\ {A}_b=54c{m}^2 \end{gathered}$$

Similarly, we need to find the area of the smaller right triangle, which is A_s, and calculated as

$$\begin{gathered} Area=\frac{1}{2}\times base\times height \\ {A}_s=\frac{1}{2}\times 3cm\times 4cm \\ {A}_s=\frac{1}{\cancel{2}}\times 3cm\times {}^2\cancel{4}cm \\ {A}_s=3cm\times 2cm \\ {A}_s=6c{m}^2 \end{gathered}$$

The ratio of the area of the bigger right triangle A_b to that of the smaller right triangle A_s is

$$\begin{gathered} A_b:A_s=54c{m}^2:6c{m}^2 \\ {A}_b:A_s=\frac{54c{m}^2}{6c{m}^2} \\ {A}_b:A_s=\frac{{}^9\cancel{54}\cancel{c{m}^2}}{{}^1\cancel{6}\cancel{c{m}^2}} \\ {A}_b:A_s=\frac{9}{1} \\ {A}_b:A_s=\mathbf{9}\mathbf{:}\mathbf{1} \end{gathered}$$

A right triangle has dimensions 11 cm by 15.6 cm by 11 cm. What type of right triangle is this? Find the perimeter of the right triangle.

Solution:

From the question, since two sides of the right triangle are equal, that means it is an isosceles right triangle.

The perimeter of the right triangle is

$$\begin{gathered} Perimeter=a+b+c \\ Perimeter=11cm+11cm+15.6cm \\ Perimeter=\mathbf{37}\mathbf{.}\mathbf{6}\mathbf{}\mathit{c}\mathit{m} \end{gathered}$$