2003 AIME II problem 7 | AIME | Math for fun and glory | Khan Academy

TL;DR

Given the radii of the circles circumscribed around two triangles within a rhombus, the area of the rhombus can be calculated using a formula relating the radii to the sides of the triangles.

Transcript

Find the area of rhombus ABCD given that the radii of the circles circumscribed about triangles ABD and ACD are 12.5 and 25, respectively. So let's draw ourselves rhombus ABCD. So let's draw a rhombus. So let me draw it. Here we go. That's a decent rhombus right over there. We know that all the sides of a rhombus are equal. And let's label the vert... Read More

Key Insights

• 🙃 A rhombus has all sides of equal length.
• ⭕ The radii of the circles circumscribed around the triangles within the rhombus can be used to find the area of the rhombus.
• 🔺 The formula relating the radius of a triangle's circumcircle to the sides of the triangle can be applied to find the area of the rhombus.
• 🙃 By solving the equations derived from the formula, the values of the sides of the rhombus can be determined.
• 🙃 The area of the rhombus can be calculated by multiplying the lengths of the sides together.

Questions & Answers

Q: What is the formula used to find the area of the rhombus?

The formula states that the radius of a triangle's circumcircle is equal to the product of the triangle's sides divided by 4 times the area of the triangle.

Q: How can the formula be applied to find the area of the rhombus?

By using the given information about the radii of the circles and relationships between the sides of the triangles, one can solve the equations derived from the formula to find the values of the sides of the rhombus.

Q: What are the radii given in the problem statement?

The radii are 12.5 and 25 for the circles circumscribed around triangles ABD and ACD, respectively.

Q: How are the sides of the rhombus related to the sides of the triangles?

The sides of the rhombus are equal to the diagonals of the triangles, which can be found using the Pythagorean theorem.

Summary & Key Takeaways

• The content explains how to find the area of a rhombus when given the radii of the circles circumscribed around two triangles within the rhombus.

• Formula: The formula used is the radius of the triangle's circumcircle is equal to the product of the triangle's sides divided by 4 times the area of the triangle.

• Applying the formula: By applying the formula to the given information about the radii and relationships between the sides of the triangles, the area of the rhombus can be calculated.