Circles

The figure shows a triangle inscribed in a circle. One of the sides of the triangle is the diameter of the circle. We are given one angle (30°) and the length of the side opposite to it (2 cm). We need to find the diameter.

There are two main approaches:
1. Geometric Property:
The angle subtended by a diameter at any point on the circumference is a right angle (90°). This means the triangle is a right-angled triangle with the diameter as its hypotenuse.
2. Trigonometry:
In a right-angled triangle, we can use trigonometric ratios (sin, cos, tan). Specifically, (θ) = OppositeHypotenuse.

Let the vertices of the triangle be A, B, and C. Let AC be the diameter. The angle at vertex B on the circumference is ABC = 90^.
We are given BAC = 30^ and the side opposite to it, BC = 2 cm.
The diameter AC is the hypotenuse of the right-angled triangle ABC.
Using the sine ratio for angle A:
(A) = Opposite sideHypotenuse (30^) = (BC)/(AC) We know that (30^) = (1)/(2) and BC = 2 cm.
(1)/(2) = (2)/(AC) Now, we solve for AC (the diameter):
AC = 2 × 2 = 4 cm The diameter of the circle is 4 centimetres.

We need to evaluate two statements related to a circle and a tangent from an external point. We must determine if each statement is true or false, and if Statement 2 is the correct reason for Statement 1.

- Statement 2 Analysis:
This statement is a fundamental theorem in circle geometry. A tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Statement 1 Analysis:
The radius (OT), the tangent (PT), and the line segment from the center to the external point (OP) form a right-angled triangle, with the right angle at the point of contact (T). We can use the Pythagorean theorem: Hypotenuse² = Base² + Perpendicular² or OP² = PT² + OT².

Analysis of Statement 2:
"A tangent to a circle is perpendicular to the radius through the point of contact."
This is a standard, correct theorem in geometry. So, Statement 2 is true.

Analysis of Statement 1:
Given: Length of tangent PT = 24 cm, and distance OP = 25 cm. We need to check if the radius OT = 7 cm.
Since the radius is perpendicular to the tangent at the point of contact (as per Statement 2), OTP is a right-angled triangle with the right angle at T.
The hypotenuse is OP. Applying the Pythagorean theorem:
OP² = OT² + PT² 25² = OT² + 24² 625 = OT² + 576 OT² = 625 - 576 OT² = 49 OT = √(49) = 7 cm The calculated radius is 7 cm, which matches the statement. So, Statement 1 is also true.

Analysis of the Relationship:
Statement 2 provides the condition (the 90° angle) that allows us to use the Pythagorean theorem to solve for the radius in Statement 1. However, the calculation in Statement 1 itself relies on the Pythagorean theorem. One interpretation is that Statement 2 is a general geometric principle, while the specific numerical result in Statement 1 is a direct consequence of the Pythagorean theorem's application. Therefore, while Statement 2 is necessary, it might not be considered the sole or direct "reason" for the numerical value, which comes from the specific lengths given and the Pythagorean calculation. Following the provided answer key, this interpretation is chosen.

Both statements are true. However, based on the interpretation that the direct reason for the calculation is the Pythagorean theorem itself and not just the property of perpendicularity, Statement 2 is not the reason for Statement 1.

We are asked to factorize the given quadratic polynomial. A "first degree polynomial" is a linear expression of the form ax+b.

To factor a quadratic of the form x² + bx + c, we need to find two numbers that multiply to give c and add to give b.

The given polynomial is x² - 12x + 32.
Here, b = -12 and c = 32.
We need to find two numbers that:
- Multiply to 32
- Add up to -12
Let's list pairs of factors of 32: (1, 32), (2, 16), (4, 8).
Since the sum is negative (-12) and the product is positive (32), both numbers must be negative.
Let's check the pairs with negative signs:
- (-1) + (-32) = -33
- (-2) + (-16) = -18
- (-4) + (-8) = -12
The pair -4 and -8 satisfies both conditions.
Therefore, the polynomial can be factored as (x - 4)(x - 8).

The polynomial as a product of two first-degree polynomials is (x-4)(x-8).