Measuring Space: Perimeter and Area | EOT

Question 14

Show that the area of a kite is half the product of its diagonals. Show this: (i) using algebra, and (ii) using geometry.

Solution

Understand the Question

A kite is a quadrilateral with two distinct pairs of equal-length adjacent sides.
A fundamental property of any kite is that its diagonals are perpendicular to each other ( $d_1 \perp d_2$ ).
We need to prove that: $\text{Area of a kite} = \dfrac{1}{2} \times d_1 \times d_2$ where $d_1$ and $d_2$ are the lengths of the two diagonals.

(i) Show that the area of a kite is half the product of its diagonals using algebra.

Step 1 · Set up the diagonals and triangle areas

Let kite $ABCD$ have diagonals $AC = d_1$ and $BD = d_2$ intersecting perpendicularly at point $O$ .Diagonal $AC$ divides kite $ABCD$ into two triangles: $\Delta ABC$ and $\Delta ADC$ .

Since $AC \perp BD$ , the altitudes of $\Delta ABC$ and $\Delta ADC$ with base $AC$ are $OB$ and $OD$ respectively.

Step 2 · Express total area algebraically

The total area of the kite is the sum of the areas of $\Delta ABC$ and $\Delta ADC$ :

\begin{aligned} \text{Area}(ABCD) &= \text{Area}(\Delta ABC) + \text{Area}(\Delta ADC) \\[0.6em] &= \left(\dfrac{1}{2} \times AC \times OB\right) + \left(\dfrac{1}{2} \times AC \times OD\right) \\[0.6em] &= \dfrac{1}{2} \times AC \times (OB + OD) \end{aligned}

Since $OB + OD = BD = d_2$ and $AC = d_1$ :

\text{Area}(ABCD) = \dfrac{1}{2} \times d_1 \times d_2

Answer

(i) $\text{Area} = \dfrac{1}{2} d_1 d_2$

(ii) Show that the area of a kite is half the product of its diagonals using geometry.

Step 1 · Enclose the kite in a bounding rectangle

Draw lines through vertices $A, B, C,$ and $D$ parallel to the diagonals $AC$ and $BD$ to form a surrounding rectangle $PQRS$ .The length and breadth of rectangle $PQRS$ are equal to the diagonals of the kite: $\text{Length} = AC = d_1, \quad \text{Breadth} = BD = d_2$

$\text{Area}(PQRS) = d_1 \times d_2$

Step 2 · Compare areas of kite and rectangle

The perpendicular diagonals divide the kite into $4$ right-angled triangles.

Rectangle $PQRS$ is composed of $8$ right-angled triangles of identical size:

$4$ triangles lie inside the kite.
$4$ triangles lie outside the kite.

Therefore, the area of the kite is exactly half the area of rectangle $PQRS$ :

\begin{aligned} \text{Area of kite} &= \dfrac{1}{2} \times \text{Area}(PQRS) \\[0.6em] &= \dfrac{1}{2} \times d_1 \times d_2 \end{aligned}

Answer

(ii) $\text{Area} = \dfrac{1}{2} d_1 d_2$

Common Mistakes

Assuming Diagonals Bisect Each Other: In a general kite, only one diagonal bisects the other, not both (unlike a rhombus).
Forgetting Perpendicularity: The formula $\frac{1}{2} d_1 d_2$ relies critically on the diagonals being perpendicular ( $90^\circ$ ). It does not apply to general quadrilaterals where diagonals meet at other angles.

More questions in EOT

Identities in algebra can sometimes be shown as area relationships. For example:

The figure shown corresponds to the identity $(a + b)^2 = a^2 + 2ab + b^2$

Do you see how?

Draw figures corresponding to the identities $(a + b)(a - b) = a^2 - b^2$ and $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$ .

An isosceles triangle has perimeter $40\text{ cm}$ ; the equal sides are $15\text{ cm}$ each. Find the area of the triangle.

An isosceles triangle has base $10\text{ cm}$ , and its area is $60\text{ cm}^2$ . What are the lengths of the equal sides?

The area of a right-angled triangle is $54\text{ sq. cm}$ . One of its legs has length $12\text{ cm}$ . Find its perimeter.

The sides of a triangle are in the ratio $2 : 3 : 4$ , and its perimeter is $45\text{ cm}$ . Find its area.

The sides of a triangle have lengths $7\text{ cm}$ , $24\text{ cm}$ , $25\text{ cm}$ . Find the area of the triangle in two different ways.

If the wheel of a bicycle has a diameter of $60\text{ cm}$ , find how far a cyclist will have travelled after the wheel has rotated $100$ times.

Find the area of a quadrant of a circle whose circumference is $66 \text{ cm}$ .

The wheel of a car has an outer radius of $28\text{ cm}$ . Calculate how far the car travels after one complete turn of the wheel, and how many times the wheel turns during a journey of $1\text{ km}$ .

Q10

Two rectangles have the same area and the same perimeter. Does this mean that they are congruent to each other?

Q11

You know that the area of a parallelogram is base $\times$ height. Using this and the figure, show that the area of a trapezium is half the sum of the parallel sides $\times$ height, i.e., $\dfrac{1}{2}(a + b)h$ .

Q12

By dividing a trapezium into two triangles show that its area is, half the sum of the parallel sides multiplied by the height (the same formula as the one given above).

Q13

Show how we can use two identical copies of a trapezium to make a parallelogram. How will this give us the formula for the area of a trapezium?

Q14

Show that the area of a kite is half the product of its diagonals. Show this: (i) using algebra, and (ii) using geometry.

Q15

Three problems about fitting congruent shapes together:

(i) Rectangle ABCD has sides $a$ , $b$ , and rectangle PQRS has sides $2a$ , $2b$ . Show that PQRS has 4 times the area of ABCD. Does this mean that 4 copies of rectangle ABCD will fit into rectangle PQRS? Check and see!

(ii) $\Delta ABC$ has sides $a$ , $b$ , $c$ , and $\Delta PQR$ has sides $2a$ , $2b$ , $2c$ . Show that $\Delta PQR$ has 4 times the area of $\Delta ABC$ . Does this mean that 4 copies of $\Delta ABC$ will fit into $\Delta PQR$ ? Check and see!

(iii) $\Delta ABC$ has sides $a$ , $b$ , $c$ , and $\Delta PQR$ has sides $3a$ , $3b$ , $3c$ . Show that $\Delta PQR$ has 9 times the area of $\Delta ABC$ . Does this mean that 9 copies of $\Delta ABC$ will fit into $\Delta PQR$ ? Check and see!

Q16

Find the fraction of the shaded region in each of the following figures:

(i) Fig. 6.43: What fraction of the triangle is shaded? (ii) Fig. 6.44: What fraction of the square is shaded?

Q17

Find the fraction of the rectangle covered by the circles in each of the following figures:

(i) Fig. 6.45: What fraction of the rectangle is covered by the circles?

(ii) Fig. 6.46: What fraction of the rectangle is covered by the circles?

Q18

Use the above to make a conjecture about the area occupied by circles fitted into a rectangle in the manner shown. Test your conjecture for particular cases: 10 circles; 20 circles; 50 circles. Then prove your conjecture!

Q19

*19. The figure shows nine identical rectangles fitted together to make a large rectangle whose area is $72\text{ cm}^2$ . Find the perimeter of each small rectangle.

Q20

Show that the areas of the shaded blue triangle and the shaded red triangle are equal.

Find a way of cutting up the blue triangle into some number of pieces and rearranging the pieces to cover the red triangle.

Q21

The figure shows a quarter circle in a square. Its centre is at one vertex, and it passes through two adjacent vertices. There are two semicircles on two adjacent sides as diameters. They create the shaded regions $A$ and $B$ .

Show that $A$ and $B$ have equal area.

Q22

In Fig. 6.50, four semicircles have been drawn within the given square whose side is $2 \text{ units}$ . The centres of these semicircles are the midpoints of the sides. They create a 4-petalled flower (shown in blue). Find the perimeter and the area of this flower.

Q23

In Fig. 6.51 we see two concentric circles with a common centre $O$ . A chord $BC$ of the larger circle is drawn, touching the smaller circle at $A$ . The length of $BC$ is $l$ . Show that the area of the green region enclosed between the two circles is $\dfrac{1}{4} \pi l^2$ .

Q24

In Fig. 6.52, semicircles have been drawn on all the sides of a right-angled triangle as shown. Show that $\text{Area (A)} + \text{Area (B)} = \text{Area (C)}$ .

Q25

Fig. 6.53 shows two circles passing through each other's centres. Find the area of the region enclosed by the two circles in terms of the common radius $r$ .

Q26

In Fig. 6.54, we see three triangles within a rectangle. The areas of the triangles are $A, B, C$ , as marked. Show that the area of the rectangle is

$\dfrac{2(A + C)(B + C)}{C}$

Q27

In the figure we see two shaded regions formed by a quarter circle, a semicircle, and a triangle.

Show that the areas of the two shaded regions are equal.