Grades 9-10 Video Solutions 2025-2025_9-10

Video Transcription

Problem 10. A circle with center O and radius 10 centimeters is given. A square OPQR is drawn inside the circle where Q is a point on the circle. What is the area of the shaded  triangle PQR? So first we'll look at this diagram and try to figure out as much as we can about the square OPQR. So we know that OQ is a diagonal of the square. And because  O is the center of the circle and Q is a point on the circle, this also means that OQ is a radius of the circle. And so OQ has length 10. So we know that OPQR is a square with  diagonal 10. This means that the length of PR is also 10. Now we can look at the triangle PQR, of which we wish to find the area. Since OPQR is a square, we know that the angle  at Q is a right angle. And so using the Pythagorean theorem, we know that the length of PQ squared  plus the length of QR squared is 10 squared. Also, since PQ and QR are both side lengths of a square, we have that PQ is equal to QR. And so if we solve this, we have that  2PQ squared is 10 squared. And this tells us that PQ is the square root of 50. QR is also the square root of 50. Now to find the area of the triangle PQR, the area of a right  triangle is the product of the legs divided by 2. And so the area of the triangle is the square root of 50 squared divided by 2, which is 25.

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