Question

(a) The length of a diagonal of a quadrilateral is 24 cm. The lengths of the perpendiculars from the
opposite vertices on this diagonal are 8.4 cm and 7.6 cm. Find the area of the quadrilateral.
Also find the ratio of the areas of the two triangles into which the diagonal divides the
quadrilateral​

Answer 1

Answer:

What is given ?

• The length of the diagonal of the quadrilateral = 24 cm.
• The height from one vertex on the diagonal is 8.4 cm.
• The height from the other vertex on the diagonal is 7.6 cm.

What we need to find ?

• Area of the quadrilateral ?
• The ratio of the areas of both the triangles into which the diagonal divides ?

Formula ?

➜ Area of Triangle = 1/2 × b × h

Solution:-

• Finding the Area of the quadrilateral.

➜ Area of Triangle 1 + Area of Triangle 2

➜ (1/2 × 24 × 8.4) + (1/2 × 24 × 7.6)

➜ 1/2 × 24 × (8.4 + 7.6)

➜ 12 × 16

➜ 192 cm²

So, the area of the quadrilateral is 192 cm².

• Finding the ratio of the areas of two triangles into which the diagonal has divided it.

➜ (1/2 × 24 × 8.4) ÷ (1/2 × 24 × 7.6)

➜ (8.4) ÷ (7.6)

➜ (42) ÷ (38)

➜ (21) ÷ (19)

So, the ratio between the areas of both the triangles is 21 by 19.

$\rule{200}2$

Answer 2

Answer:

Area of quadrilateral ABCD = 192cm²

ratio = 21/19

Step-by-step explanation:

Let us consider a quadrilateral ABCD

The diagonal be AC

and two perpendiculars BE and DF are drawn from the opposite vertices B and D .

Given ,

BE = 7.6cm

DF = 8.4cm

and diagonal , AC = 24cm

We know that ,

Area of triangle = 1/2 ×base × height

$ar.(ADC) = \frac{1}{2} \times 8.4 \times 24 \\ \implies ar.(ADC) = 8.4 \times 12 \\ \implies ar.(ADC) = 100.8 {cm}^{2}$

And again

$ar.(ABC) = \frac{1}{2} \times 7.6 \times 24 \\ \implies ar.(ABC) = 12 \times 7.6 \\ \implies ar.(ABC) = 91.2 {cm}^{2}$

Now ,

Area of quadrilateral ABCD = Sum of areas of the triangles

$ar.(ABCD) = ar.(ADC) + ar.(ABC) \\ \implies ar.(ABCD) = 100.8 {cm}^{2} + 91.2 {cm}^{2} \\ \implies ar.(ABCD) = 192 {cm}^{2}$

Now the ratio of the areas the triangles are :

$\frac{ar.(ADC)}{ar(ABC)} = \frac{8.4}{7.6} \\ \implies \frac{ar.(ADC)}{ar.(ABC)} = \frac{21}{19}$