The above diagram is a regular pentagram. Let \(\lvert\triangle TRS\rvert\) denote the area of \(\triangle TRS,\) and \(\lvert\square PQRS\rvert\) the area of \(\square PQRS.\) If \(\lvert\triangle ABC\rvert=52,\) what is the area of the shaded region \[\lvert\square PQRS\rvert - \lvert\triangle TRS\rvert?\]

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations,
examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Your answer seems reasonable.
Find out if you're right!

In the above diagram, \(\square ABCD\) is a square and \(\overline{DE}\) is perpendicular to one of its diagonals \(\overline{AC}.\) If the length of \(\overline{DE}\) is \(21.\) what is the length of \(\overline{AC}?\)

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations,
examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Your answer seems reasonable.
Find out if you're right!

In the above diagram, figure \(ABCDE\) is a regular pentagon centered at point \(F\). Find the area of quadrilateral \(CDEF\) when the area of triangle \(ABF\) is \(56.\)

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations,
examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Your answer seems reasonable.
Find out if you're right!

In \(\triangle ABC\) above, \(L,M\) and \(N\) are the midpoints of \(\overline {AB}, \overline {BC}\) and \(\overline {AC},\) respectively. If the area of \(\triangle LMN\) is \(54,\) what is the area of \(\triangle ABC?\)

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations,
examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Your answer seems reasonable.
Find out if you're right!