Calculation of length of the height

Find the value of t in the following right-angled triangle:

Solution

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

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

From the figure, $a = 8$ cm, $b = t$ and $c = 17$ cm.

Thus, $t^2 = (17 \text{ cm})^2 - (8 \text{ cm})^2$ $t^2 = 289 \text{ cm}^2 - 64 \text{ cm}^2$ $t^2 = 225 \text{ cm}^2$.

Find the square root on both sides: $\sqrt{t^2} = \sqrt{225 \text{ cm}^2}$ $t = 15$ cm.

Therefore, the value of t is 15 cm.

Find the value of h in the following right-angled triangle:

Solution

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

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

From the figure, $a = 9$ cm, $b = h$ and $c = 15$ cm.

Thus, $h^2 = (15 \text{ cm})^2 - (9 \text{ cm})^2$ $h^2 = 225 \text{ cm}^2 - 81 \text{ cm}^2$ $h^2 = 144 \text{ cm}^2$.

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

Therefore, the value of h is 12 cm.

Find the value of d in the following right-angled triangle:

Solution

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

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

From the figure, $a = 12$ cm, $b = d$ and $c = 20$ cm.

Thus, $d^2 = (20 \text{ cm})^2 - (12 \text{ cm})^2$ $d^2 = 400 \text{ cm}^2 - 144 \text{ cm}^2$ $d^2 = 256 \text{ cm}^2$.

Find the square root on both sides: $\sqrt{d^2} = \sqrt{256 \text{ cm}^2}$ $d = 16$ cm.

Therefore, the value of d is 16 cm.

A ladder 13 metres long is placed against a vertical house wall. If the length between the foot of the ladder to the wall is 5 metres, find the height of the wall from the floor to the ladder.

Solution

Use the figure below:

The distance between the wall and the foot of the ladder is the base, which is $a = 5$ m.

The length of the ladder is the hypotenuse, which is $c = 13$ m.

The length of the height is $b = h$.

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

From the figure, $a = 5$ m, $b = h$ and $c = 13$ m.

Thus, $h^2 = (13 \text{ m})^2 - (5 \text{ m})^2$ $h^2 = 169 \text{ m}^2 - 25 \text{ m}^2$ $h^2 = 144 \text{ m}^2$.

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

Therefore, the height of the wall is 12 metres.