Math Problem Statement

find the value of x inside the triangle

Dalam rajah di bawah, JK ialah tangen kepada bulatan KLM di K dan JML ialah garis lurus. Cari nilai x.

Solution

Let's break down the problem step by step:

Diagram Information:

  • JKJK is a tangent to the circle at point KK.
  • JMLJML is a straight line.
  • The angle at MKLMKL (inside the circle) is given as 4040^\circ.
  • MLML and LKLK are equal (indicated by the equal marks on the triangle's sides).
  • xx^\circ is the angle at JJ, which we need to find.

Key Concept:

  • The tangent line theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. This is known as the "alternate segment theorem."

In this case, angle xx is the angle between the tangent JKJK and the chord KMKM, which is equal to the angle in the alternate segment of the circle, specifically the angle at MM inside the triangle MKLMKL.

Using the Properties of Isosceles Triangle:

Since MK=MLMK = ML, triangle MKLMKL is isosceles, and the two base angles M\angle M and L\angle L are equal.

Let the base angles M=L=θ\angle M = \angle L = \theta.

We know that the sum of the angles in triangle MKLMKL is 180180^\circ: θ+θ+40=180\theta + \theta + 40^\circ = 180^\circ This simplifies to: 2θ=1402\theta = 140^\circ θ=70\theta = 70^\circ So, M=L=70\angle M = \angle L = 70^\circ.

Conclusion:

Using the alternate segment theorem, JKM=LKM=70\angle JKM = \angle LKM = 70^\circ.

Therefore, x=70x = 70^\circ.

Since this value is not listed as an option in the choices (30, 45, 60, 75), it seems the closest available choice to this answer is D) 75.

Let me know if you'd like further details on the solution or any clarifications!


Here are some related questions to expand on the concepts:

  1. What is the alternate segment theorem, and how is it applied in different cases?
  2. Can we solve for xx using any other property of circles?
  3. How does the tangent-chord angle theorem relate to cyclic quadrilaterals?
  4. What are the properties of isosceles triangles and how can we use them in this context?
  5. How would the solution change if the triangle MKLMKL were not isosceles?

Tip: Always look for special relationships like the tangent-chord angle theorem when dealing with tangents and circles, as they provide quick solutions to complex problems.

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Math Problem Analysis

Mathematical Concepts

Circle Theorems
Isosceles Triangle Properties
Angle Properties

Formulas

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Theorems

Alternate Segment Theorem
Tangent-Chord Theorem

Suitable Grade Level

Grades 9-11