Video: Finding the Length of a Side in a Right-Angled Triangle Using the Pythagorean Theorem

Circle 𝑀 has radius 11 cm. If 𝐢𝐴 = 16.3 cm, what is 𝐴𝐡? Answer to the nearest tenth.

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Video Transcript

Circle 𝑀 has radius 11 centimeters. If 𝐢𝐴 is equal to 16.3 centimeters, what is 𝐴𝐡? Answer to the nearest tenth.

So we’re given a diagram of a circle and then a triangle formed by connecting three points: 𝑀, which is the center of the circle; 𝐡, which is a point on the circumference; and 𝐴, which is an external point. We’re asked to calculate the length of the line segment 𝐴𝐡. We’re given various pieces of information in the question. Firstly, that the length of the line 𝐢𝐴 is equal to 16.3 centimeters. Secondly, we’re told that the radius of the circle is equal to 11 centimeters. Remember, the radius of a circle is a line whose end points are the center of the circle and the point on the circumference. So in fact, there are two radii in the diagram, the lines 𝑀𝐡 and 𝑀𝐢.

Having added these labels to the diagram, we can now see that the length we’ve been asked to calculate, 𝐴𝐡, is in fact the third side of a triangle, triangle 𝐴𝑀𝐡. And we know the length of the other two sides of the triangle. 𝐡𝑀, remember, is a radius of the circle. So it’s 11 centimeters. And the full length of the side 𝐴𝑀 is 27.3 centimeters. Do we know anything else about this triangle? Well, in fact we do. 𝐴𝐡 is a tangent to the circle. And a key fact about tangents to circles is this: a tangent to a circle is perpendicular to the radius at the point of contact. The radius at this point is the line 𝑀𝐡. And so we know that the lines 𝑀𝐡 and 𝐴𝐡 are perpendicular. Therefore, this triangle is a right-angled triangle.

If we know the length of two sides of a right-angled triangle and we want to calculate the length of the third side, we can do this by applying the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In our triangle, this means that 𝐴𝐡 squared plus 11 squared is equal to 27.3 squared. This gives an equation that we can solve in order to find the length of 𝐴𝐡.

The first step is to evaluate both 11 squared and 27.3 squared. This gives 𝐴𝐡 squared plus 121 is equal to 745.29. The next step is to subtract 121 from each side of the equation. We now have 𝐴𝐡 squared is equal to 624.29. To find the value of 𝐴𝐡, we need to take the square root of both sides. This gives 𝐴𝐡 is equal to the square root of 624.29, which as a decimal is equal to 24.98579.

Remember, the question asked as to give our answer to the nearest tenth. So we need to round this decimal. Our answer to the problem is that the length of 𝐴𝐡, to the nearest tenth, is 25.0 centimeters. Remember, the key fact we used in this question was that a tangent to a circle is perpendicular to the radius at the point of contact.

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