Math Problem Statement

TA is a tangent to the circle ABCD at the point A. It is given that angle BAD = 88°, angle ABD = 46°, and BC = CD. Prove that AC is the diameter of the circle. Also, find the angle DAT.

Solution

Let's break down the problem step by step:

Given Information:

  • TATA is a tangent to the circle at point AA.
  • BAD=88\angle BAD = 88^\circ
  • ABD=46\angle ABD = 46^\circ
  • BC=CDBC = CD

The problem asks:

(a) Prove that ACAC is the diameter of the circle.

(b) Find DAT\angle DAT.


(a) Proving ACAC is the diameter of the circle:

Step 1: Using the tangent property at AA

Since TATA is a tangent to the circle at point AA, the angle between the tangent and the chord ADAD, which is DAT\angle DAT, must be equal to the angle subtended by the chord ADAD on the opposite side of the circle. In other words: DAT=ACD\angle DAT = \angle ACD

Step 2: Using the cyclic quadrilateral property

ABCDABCD forms a cyclic quadrilateral (a quadrilateral inscribed in a circle). In a cyclic quadrilateral, opposite angles sum up to 180180^\circ. Thus: BAD+BCD=180\angle BAD + \angle BCD = 180^\circ Substituting the given value for BAD=88\angle BAD = 88^\circ, we get: 88+BCD=18088^\circ + \angle BCD = 180^\circ BCD=92\angle BCD = 92^\circ

Step 3: Using the isosceles triangle property

Since BC=CDBC = CD, triangle BCDBCD is isosceles. Therefore, the base angles BDC\angle BDC and BCD\angle BCD are equal. Let these angles be xx. Thus: 2x+92=1802x + 92^\circ = 180^\circ Solving for xx: 2x=88x=442x = 88^\circ \quad \Rightarrow \quad x = 44^\circ So, BCD=44\angle BCD = 44^\circ and BDC=44\angle BDC = 44^\circ.

Step 4: Angle subtended by the diameter

In a circle, the angle subtended by the diameter at the circumference is always 9090^\circ. Since ACD=90\angle ACD = 90^\circ, it proves that ACAC must be the diameter of the circle.


(b) Finding DAT\angle DAT

From the tangent-secant theorem, we know that: DAT=ACD\angle DAT = \angle ACD From the cyclic quadrilateral property and previous calculations, we already found that ACD=44\angle ACD = 44^\circ.

Thus, DAT=44\angle DAT = 44^\circ.


Would you like more details or further explanation? Here are some follow-up questions to expand your understanding:

  1. Why is the tangent-secant theorem important in this context?
  2. How does the property of cyclic quadrilaterals help in angle calculations?
  3. What would change if the circle were not cyclic?
  4. How can we verify that ACAC is indeed the diameter geometrically?
  5. What would be the next step if the triangle were not isosceles?

Tip: Always verify relationships between angles in cyclic quadrilaterals using the sum of opposite angles property to avoid calculation errors.

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Math Problem Analysis

Mathematical Concepts

Circle Geometry
Cyclic Quadrilaterals
Tangent-Secant Theorem
Isosceles Triangles

Formulas

Angle sum of cyclic quadrilateral: Opposite angles sum to 180°
Tangent-Secant Theorem: Angle between tangent and chord is equal to angle subtended by the chord
Properties of an isosceles triangle: Base angles are equal

Theorems

Tangent-Secant Theorem
Cyclic Quadrilateral Theorem
Isosceles Triangle Property
Angle subtended by a diameter is 90°

Suitable Grade Level

Grade 10-12