Question

Each side of a square ABCD has a length of 1 unit. Points P and Q belong to AB and DA respectively. The perimeter of APQ is 2 units. What will be angle of PCQ? Please give the shortest solution, without trigonometry if possible.

Answer 1

The angle PCQ in triangle APQ is 0 degrees.

Step-by-step explanation:

Step 1: Let's draw a square ABCD with each side having a length of 1 unit.

Step 2: Let P be a point on AB and Q be a point on DA.

Step 3: The perimeter of the triangle APQ is given as 2 units. Since the side length of the square is 1 unit, the length of AP and AQ will be 1 unit each.

Step 4: The length of PQ can be found using the perimeter formula for a triangle: perimeter = side1 + side2 + side3. In this case, 2 = AP + AQ + PQ. So, PQ = 2 - 1 - 1 = 0 units.

Step 5: The angle PCQ can be found using the law of cosines: cos(PCQ) = (PQ^2 + PC^2 - CQ^2) / (2 * PQ * PC). Since PQ = 0 units, the formula simplifies to cos(PCQ) = PC / CQ.

Step 6: Since the side length of the square is 1 unit, PC and CQ are both equal to 1 unit. Therefore, cos(PCQ) = 1 / 1 = 1.

Step 7: The angle PCQ can be found by taking the inverse cosine of 1, which is 0 degrees.

So, the angle PCQ is 0 degrees.