one diagonal of a cyclic quadrilateral coincides with a diameter of a circle whose area is 36pi cm^2. If the other diagonal which measures 8cm meets the first diagonal at right angles, find the area of quadrilateral. Show your solutions
The diameter of the circle is 12 cm.
The quadrilateral consists of two congruent triangles, with base 12 and altitude 4. So, its area is 48 cm^2
How did you got your answer. I need a complete solution please.
To find the area of the quadrilateral, we need to determine the length of the diagonal that coincides with the diameter of the circle.
We know that the area of the circle is 36π cm², so we can use the formula for the area of a circle, A = πr², to find the radius of the circle.
36π = πr²
Dividing both sides by π:
r² = 36
Taking the square root of both sides:
r = √36
r = 6 cm
Since the diameter of the circle coincides with one diagonal of the quadrilateral, the length of this diagonal is equal to twice the radius:
Length of diagonal = 2 × 6 = 12 cm
Now, we are given that the other diagonal measures 8 cm and meets the first diagonal at right angles. We can use the lengths of the diagonals to split the quadrilateral into four right-angled triangles.
To find the area of the quadrilateral, we need to calculate the sum of the areas of these four triangles.
Each triangle has a base of 8 cm and a height of 6 cm (half the length of the diagonal coinciding with the diameter).
Area of one triangle = (1/2) × base × height = (1/2) × 8 × 6 = 24 cm²
Since there are four triangles in the quadrilateral, the total area of the quadrilateral is:
Area of quadrilateral = 4 × Area of one triangle = 4 × 24 cm² = 96 cm²
Therefore, the area of the quadrilateral is 96 cm².
To find the area of the cyclic quadrilateral, we need to determine its side lengths using the given information and then apply the appropriate formula. Let's break down the solution step by step:
1. Start by drawing a diagram of the cyclic quadrilateral. Label the points where the diagonals intersect as A and C, and let O represent the center of the circle.
C
|
|
O
/ \
/ \
/ \
/ \
---------A---------
2. We know that one diagonal of the cyclic quadrilateral coincides with a diameter of the circle. Let's assume that AC is the diameter and that its length is 2r, where r is the radius of the circle. Therefore, AC = 2r.
3. The area of the circle is given as 36π cm^2. We can utilize the formula for the area of a circle, A = πr^2, where A represents the area and r is the radius.
36π = πr^2
Dividing both sides of the equation by π, we get:
36 = r^2
Taking the square root of both sides, we find:
r = √36
r = 6 cm
So, the radius of the circle is 6 cm.
4. Since one diagonal, AC, coincides with the diameter of the circle, it is equal to 2r:
AC = 2(6) = 12 cm
5. We are given that the other diagonal, BD, measures 8 cm and meets AC at right angles. From this information, we can infer that the diagonals bisect each other at right angles, and the quadrilateral is a rectangle. Therefore, the length of the other diagonal, BD, is also 12 cm.
6. We have now determined the side lengths of the rectangle, AC and BD, both measuring 12 cm. The formula to find the area of a rectangle is A = length x width. In this case, the length and width of the rectangle are both 12 cm.
Area of the rectangle = 12 cm x 12 cm = 144 cm^2
7. Thus, the area of the given cyclic quadrilateral is 144 square cm.