Calculation of length of the base

Calculation of length of the base

The length of the base of the right-angled triangle can be found if the lengths of height and hypotenuse are known. If the length of the base is '$a$', length of the height is '$b$', and length of hypotenuse is '$c$', apply the Pythagoras' theorem $a^2 + b^2 = c^2$ to calculate the length of base '$a$' as follows:

Subtract $b^2$ from both sides: $a^2 + b^2 - b^2 = c^2 - b^2$ $a^2 = c^2 - b^2.$

Find the square root on both sides: $\sqrt{a^2} = \sqrt{c^2 - b^2}$ $a = \sqrt{c^2 - b^2}.$

Therefore, the length of the base is equal to the square root of the difference between the squares of the length of hypotenuse and height.

Example 1

Find the value of $y$ in the following right-angled triangle:

Solution

Using the Pythagoras' theorem: $a^2 + b^2 = c^2.$

Subtract $b^2$ from both sides: $a^2 + b^2 - b^2 = c^2 - b^2$ $a^2 = c^2 - b^2.$

From the figure, $a = y$, $c = 17$ cm and $b = 15$ cm

Thus, $y^2 = (17\text{ cm})^2 - (15\text{ cm})^2$ $y^2 = 289\text{ cm}^2 - 225\text{ cm}^2$ $y^2 = 64\text{ cm}^2.$

Find the square root on both sides: $\sqrt{y^2} = \sqrt{64\text{ cm}^2}$ $y = 8\text{ cm}.$

Therefore, the value of $y$ is 8 cm.

Example 2

Find the value of $v$ in the following right-angled triangle:

Solution

Using the Pythagoras' theorem: $a^2 + b^2 = c^2.$

Subtract $b^2$ from both sides of the equation: $a^2 + b^2 - b^2 = c^2 - b^2$ $a^2 = c^2 - b^2.$

From the figure, $c = 20$ cm, $b = 16$ cm and $a = v$

Thus, $v^2 = (20\text{ cm})^2 - (16\text{ cm})^2$ $v^2 = 400\text{ cm}^2 - 256\text{ cm}^2$ $v^2 = 144\text{ cm}^2.$

Find the square root on both sides: $\sqrt{v^2} = \sqrt{144\text{ cm}^2}$ $v = 12\text{ cm}.$

Therefore, the value of $v$ is 12 cm.

Example 3

Find the value of $u$ in the following right-angled triangle:

Solution

Using the Pythagoras' theorem: $a^2 + b^2 = c^2.$

Subtract $b^2$ from both sides: $a^2 + b^2 - b^2 = c^2 - b^2$ $a^2 = c^2 - b^2.$

From the figure, $c = 10$ cm, $b = 8$ cm and $a = u$.

Thus, $u^2 = (10\text{ cm})^2 - (8\text{ cm})^2$ $u^2 = 100\text{ cm}^2 - 64\text{ cm}^2$ $u^2 = 36\text{ cm}^2.$

Find the square root on both sides: $\sqrt{u^2} = \sqrt{36\text{ cm}^2}$ $u = 6\text{ cm}.$

Therefore, the value of $u$ is 6 cm.