Math Problem Statement

In the figure, AC is a tangent to the circle centre O, what is the area of ΔOCA?

Solution

To solve for the area of triangle ΔOCA\Delta OCA, we will follow these steps:

Step 1: Identify given values

  • OA=14cmOA = 14 \, \text{cm} (this is the radius of the circle)
  • OC=8cmOC = 8 \, \text{cm} (another radius of the circle)
  • ACAC is a tangent at point CC to the circle, which means that OCA=90\angle OCA = 90^\circ.

Step 2: Use the Pythagorean theorem

Since ΔOCA\Delta OCA is a right-angled triangle at CC, we can apply the Pythagorean theorem to find the length of ACAC.

OA2=OC2+AC2OA^2 = OC^2 + AC^2 Substituting the values of OAOA and OCOC:

142=82+AC214^2 = 8^2 + AC^2 196=64+AC2196 = 64 + AC^2 AC2=132AC^2 = 132 AC=13211.49cmAC = \sqrt{132} \approx 11.49 \, \text{cm}

Step 3: Calculate the area of ΔOCA\Delta OCA

The area of a right-angled triangle is given by:

Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} Here, OC=8cmOC = 8 \, \text{cm} is the base and AC11.49cmAC \approx 11.49 \, \text{cm} is the height.

Area=12×8×11.49=45.96cm2\text{Area} = \frac{1}{2} \times 8 \times 11.49 = 45.96 \, \text{cm}^2

Thus, the area of ΔOCA\Delta OCA is approximately 45.96 cm².

Would you like more details or have any questions?

Here are some related questions:

  1. How do you calculate the hypotenuse in a right triangle?
  2. How does the tangent property of circles help in solving geometry problems?
  3. What is the significance of the right angle in triangle area calculations?
  4. Can you generalize the method to find the area of any right triangle?
  5. How do you use the Pythagorean theorem in three-dimensional geometry?

Tip: Always remember that a tangent to a circle forms a right angle with the radius at the point of tangency.

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Math Problem Analysis

Mathematical Concepts

Geometry
Right Triangle
Circle Theorems

Formulas

Pythagorean theorem: a² + b² = c²
Area of a triangle: (1/2) × base × height

Theorems

Pythagorean Theorem
Tangent-Radius Theorem (Tangent forms a right angle with the radius)

Suitable Grade Level

Grades 8-10