# 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 ππ΄ = 17.2 cm and π΄π΅ = 27.6 cm, find the length of the line segment ππΆ and the area of β³π΄π·π΅ to the nearest tenth.

03:24

### Video Transcript

In the figure below, if ππ΄ is equal to 17.2 centimeters and π΄π΅ is equal to 27.6 centimeters, find the length of the line segment ππΆ and the area of triangle π΄π·π΅ to the nearest tenth.

Since π is the center of the circle and the line segment ππ· bisects the chord π΄π΅ at πΆ, we can apply the chord bisector theorem. This states that if we have a circle with center π containing a chord π΄π΅, then the straight line that passes through π and bisects π΄π΅ is perpendicular to π΄π΅. This means that the measure of angle ππΆπ΄ is 90 degrees. Since the chord π΄π΅ has length 27.6 centimeters and we know that πΆ bisects this chord, π΄πΆ is equal to 27.6 divided by two. This is equal to 13.8 centimeters.

We are also told that the radius ππ΄ is equal to 17.2 centimeters. We can therefore use the Pythagorean theorem in the right triangle ππΆπ΄ such that ππΆ 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 ππΆ is equal to 10.3 centimeters.

The second part of our question asks us to calculate the area of triangle π΄π·π΅. 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 π΄π΅ is equal to 27.6 centimeters. The perpendicular height πΆπ· will be equal to the length of ππ· minus the length of ππΆ. ππ· is the radius of the circle, and we know this is equal to 17.2 centimeters. Whilst we could use 10.3 centimeters for ππΆ, it is better for accuracy to use the nonrounded version: ππΆ is equal to 10.266 and so on. Subtracting this from 17.2, we see that the length of πΆπ· is 6.933 and so on centimeters.

We can now calculate the area of triangle π΄π·π΅ 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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.