Geometry Puzzle – Critical Thinking 16
Two sides of a triangle are 10 and 5. What areas are possible? The triangle inequality forces the third side strictly between 5 and 15. The area is at its largest when the included angle between the two given sides is 90 degrees, giving a maximum area of (1/2)(10)(5) = 25 square units.
Key takeaways:
- Triangle inequality: each side must be less than the sum of the other two, so the third side here is strictly between 5 and 15.
- Area of a triangle with two known sides \( a, b \) and the included angle \( \theta \) is \( \frac{1}{2} ab \sin\theta \).
- Sine peaks at \( \theta = 90^\circ \), so the maximum area is \( \frac{1}{2}(10)(5) = 25 \).
- As \( \theta \) approaches 0 or 180 degrees, the triangle degenerates and area shrinks toward 0.
- So the possible area range for this triangle is \( 0 < A \leq 25 \) (exclusive of 0, inclusive of 25 at the right angle).
This is a creative Puzzle to challenge the Brain! This is for 5th graders and older. A detailed solution is also provided
Challenge:
Two sides of a triangle measure 10 and 5. Which of the following could be the area of the triangle?
A- 50
B- 35
C- 25
D- 2
E- 0.1
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The correct answer is C, D and E.
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We can find the maximum area of the triangle only when the given sides are placed at right angles. Why?
Let’s put 10 as the base and 5 as the height. Then, the area of the triangle is: \(\frac{10×5}{2}= 25\)
The angle less or more than 90 degrees between the sides reduce the area. (Draw it out for yourself). Hence, the area can be anything between 0 and 25.
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Frequently Asked Questions
What is a mathematical puzzle?
A mathematical puzzle is a problem with a clear goal that needs reasoning more than routine calculation. This triangle puzzle is a classic: with sides of 10 and 5, which areas are possible? The answer comes from the formula A = ½ × 10 × 5 × sin θ. The area peaks at 25 when the sides meet at 90°, so 25, 2 and 0.1 work, but 35 and 50 do not.
Are puzzles considered math?
Many puzzles are math in disguise, especially ones about shapes, numbers or logic. This geometry puzzle uses two real school topics: the triangle inequality, which keeps the third side strictly between 5 and 15, and the area of a triangle. Solving it shows why a fixed pair of sides can still make many different triangles with many different areas.
What is mathematics critical thinking?
It is the skill of reasoning through a math problem, testing ideas and justifying the answer. In this triangle puzzle, a student might first guess that the area is fixed. Critical thinking means asking what happens as the angle between the sides of 10 and 5 changes. Picture the angle opening and closing, and you see the area grow to 25 at a right angle and shrink toward 0.
What are critical thinking geometry problems?
Critical thinking geometry problems ask you to reason about shapes instead of just plugging numbers into a formula. Asking which areas are possible for a triangle with sides 10 and 5 is a good example. Treat 10 as the base: the height can be at most 5, so the area is at most ½ × 10 × 5 = 25 and can get as close to 0 as you like.
Are there math puzzles for high school geometry?
Yes. Puzzles like this one fit well in high school geometry because they combine the triangle inequality with area and, later, trigonometry. With sides 10 and 5, students can use A = ½ab sin θ to see that the largest area is 25, reached at a 90° angle. A follow-up challenge: find the third side in that case, which is √125 = 5√5 ≈ 11.18.
Related Lessons You May Like
- How to classify triangles
- How to find the area of triangles
- How to use the Pythagorean Theorem
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If your student enjoys puzzles like this, Geometry for Beginners works the same relationships inside a full curriculum. For the number-theory side, Pre-Algebra for Beginners covers primes, factors, and divisibility from the ground up.
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