Question Video: Finding the Perpendicular Distance between a Chord and the Center of a Circle and the Area of the Triangle Inside It Mathematics

In the figure below, if 𝑀𝐴 = 17.2 cm and 𝐴𝐡 = 27.6 cm, find the length of the line segment 𝑀𝐢 and the area of △𝐴𝐷𝐡 to the nearest tenth.

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Video Transcript

In the figure below, if 𝑀𝐴 is equal to 17.2 centimeters and 𝐴𝐡 is equal to 27.6 centimeters, find the length of the line segment 𝑀𝐢 and the area of triangle 𝐴𝐷𝐡 to the nearest tenth.

Since 𝑀 is the center of the circle and the line segment 𝑀𝐷 bisects the chord 𝐴𝐡 at 𝐢, we can apply the chord bisector theorem. This states that if we have a circle with center 𝑀 containing a chord 𝐴𝐡, then the straight line that passes through 𝑀 and bisects 𝐴𝐡 is perpendicular to 𝐴𝐡. This means that the measure of angle 𝑀𝐢𝐴 is 90 degrees. Since the chord 𝐴𝐡 has length 27.6 centimeters and we know that 𝐢 bisects this chord, 𝐴𝐢 is equal to 27.6 divided by two. This is equal to 13.8 centimeters.

We are also told that the radius 𝑀𝐴 is equal to 17.2 centimeters. We can therefore use the Pythagorean theorem in the right triangle 𝑀𝐢𝐴 such that 𝑀𝐢 is equal to the square root of 17.2 squared minus 13.8 squared. This is equal to 10.266 and so on. And rounding to the nearest tenth, the line segment 𝑀𝐢 is equal to 10.3 centimeters.

The second part of our question asks us to calculate the area of triangle 𝐴𝐷𝐡. Clearing some space, we recall that the area of any triangle is equal to the length of its base multiplied by the length of its perpendicular height divided by two. We know that the base of our triangle 𝐴𝐡 is equal to 27.6 centimeters. The perpendicular height 𝐢𝐷 will be equal to the length of 𝑀𝐷 minus the length of 𝑀𝐢. 𝑀𝐷 is the radius of the circle, and we know this is equal to 17.2 centimeters. Whilst we could use 10.3 centimeters for 𝑀𝐢, it is better for accuracy to use the nonrounded version: 𝑀𝐢 is equal to 10.266 and so on. Subtracting this from 17.2, we see that the length of 𝐢𝐷 is 6.933 and so on centimeters.

We can now calculate the area of triangle 𝐴𝐷𝐡 by multiplying this by 27.6 centimeters and then dividing by two. This is equal to 95.683 and so on. Once again, we need to round to the nearest tenth, giving us an answer of 95.7 square centimeters.

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