+21 Let ABTCD be a convex pentagon with area 22 such that AB = CD and the circumcircles of triangles TAB and TCD are internally tangent. Given that ∠ATD = 90◦ , ∠BTC = 120◦ , BT = 4, and CT = 5, compute the area of triangle TAD.
+9673 This is from HMMT 2024 Geometry Round, and the solution is here: https://hmmt-archive.s3.amazonaws.com/tournaments/2024/feb/geo/solutions.pdf
+743 Analyzing the Given Information:
Convex Pentagon: ABTCD is a convex pentagon, meaning all interior angles are less than 180 degrees.
Equal Sides: AB = CD implies sides AB and CD are congruent.
Internally Tangent Circles: The circumcircles of triangles TAB and TCD touch each other inside the pentagon.
Right Angle: Angle ATD is a right angle (90 degrees).
Angles and Sides: Angle BTC is 120 degrees, BT = 4 (length of side BT), and CT = 5 (length of side CT).
Key Points for Area Calculation:
Right Triangle: Triangle TAD is a right triangle due to angle ATD being 90 degrees.
Power of a Point: Since the circumcircles of TAB and TCD are internally tangent, point T is the power of point A with respect to circle BTC.
This means the product of TA and its projection onto BT (let's call it AP) is equal to the square of BT (which is 4).
Solving for AP and Area of Triangle TAD:
Projecting AP: Since triangle BTC is isosceles with angle BTC = 120 degrees, line BT bisects angle ABC. This means projection point P lies on line BT.
Using Power of a Point: Since T is the power of point A with respect to circle BTC, we have:
TA * AP = BT^2 (given)
Substitute known values: TA * AP = 4^2 = 16
Pythagorean Theorem in Triangle TAP: We know AP and need to find TA (the hypotenuse) and the area of triangle TAD (which is 1/2 * base * height).
Use the Pythagorean theorem: TA^2 = AP^2 + TP^2 (where TP is the leg opposite the right angle)
Substitute known values from step 2: TA^2 = 16 + TP^2
Finding TP: Since BT bisects angle ABC, triangle ABT is a 30-60-90 triangle with BT = 4 (half of the hypotenuse) and CT = 5 (the other leg). This implies AB = 2 * BT = 8 (hypotenuse).
Using Pythagoras in triangle ABT: TP^2 = AB^2 - BT^2 = 8^2 - 4^2 = 48
Finding TA and Area of Triangle TAD:
Substitute TP^2 from step 4 into the equation from step 3: TA^2 = 16 + 48 = 64
Take the square root of both sides to find TA: TA = 8
Now calculate the area of triangle TAD: Area = 1/2 * base * height = 1/2 * 4 (base TD) * 8 (height TA) = 16
Answer:
Therefore, the area of triangle TAD is 16.
