Question

Find the area of the following trapezium.

Answer 1

Given:

• Trapezium ABCD with AB = 25 cm, BC = 17 cm and DC = 40 cm.
• AD is perpendicular to DC right angle at D.

Step 1: Identify that AD is perpendicular to DC, so AD is the height (h) of the trapezium.

We need to calculate the height AD.

Step 2: In right triangle B C D, use Pythagoras theorem to find AD.

Since BC = 17 cm, DC = 40 cm and AD is perpendicular to DC, AD can be found using the right triangle A D C if we find the length of AD.

Step 3: Use triangle ABD right triangle where AB = 25 cm (top base) and DC = 40 cm (bottom base).

Height AD is to be calculated using triangle B C D.

Step 4: Calculate height AD.

Let height AD = h.

In triangle B C D, BD is the hypotenuse and sides are BC = 17 cm and DC = 40 cm minus AB = 25 cm the portion of DC adjacent to AB.

So, `BD = sqrt((DC - AB)^2 + BC^2)`

= `sqrt((40 - 25)^2 + 17^2)`

= `sqrt(15^2 + 17^2)`

= `sqrt(225 + 289)`

= `sqrt(514)`

However, a simpler approach is to note that AD is perpendicular to DC, so AD is the height.

Step 5: Using Pythagoras theorem in triangle ABD:

The base BD = ?

Need to calculate AD:

Given the right angle at D, triangle ADC is right angled.

Using Pythagoras theorem:

AD² + DC² = AC², but AC is not given.

Alternatively, since AB is parallel to DC, the area formula for trapezium is:

Area = `1/2`× sum of parallel sides × height

Here, parallel sides are AB = 25 cm and DC = 40 cm.

We need height AD.

Step 6: To find height AD, apply Pythagoras theorem in triangle B C D noting BD² = BC² + CD²:

Given right angle at D, triangle ADC is right angled AD² + DC² = AC².

Still missing length of AC.

Step 7: Another approach: Since AB is parallel and length 25 cm and DC is 40 cm, the difference is 15 cm.

Length BC = 17 cm.

Let the perpendicular distance (height) between the two parallel sides be h.

Now, in triangle B C D:

Using Pythagoras theorem to find height h:

`H = sqrt(BC^2 - (DC - AB)^2)`

= `sqrt(17^2 - (40 - 25)^2)`

= `sqrt(289 - 15^2)`

= `sqrt(289 - 225)`

= `sqrt(64)`

= 8 cm

Step 8: Calculate area of trapezium:

Area = `1/2`× sum of parallel sides × height

= `1/2 xx (25 + 40) xx 8`

= `1/2 xx 65 xx 8`

= 32.5 × 8

= 260 cm²

The area of the trapezium ABCD is 260 cm².