Geometric Analysis: Determining Acute Triangle Properties

Q: How can I tell if a triangle is acute without measuring angles?

Use the acute triangle test: If $ c^2 where c is the longest side, the triangle is acute. If \( c^2 = a^2 + b^2 $, it's right. If $ c^2 > a^2 + b^2 $, it's obtuse.

Q: What makes a triangle NOT acute?

A triangle is not acute if it has even one angle that is $ 90^\circ $ or greater. Having just one right angle or obtuse angle disqualifies it from being acute.

Q: Can I trust what triangles look like in drawings?

Never rely on visual appearance alone! Drawings may not be to scale or might be misleading. Always use mathematical tests or given measurements to classify triangles accurately.

Q: What's the difference between acute, right, and obtuse triangles?

Acute: All angles $ 90^\circ $Right: One angle = $ 90^\circ $Obtuse: One angle > $ 90^\circ $Remember: A triangle can only be one type!

Q: Why does the Pythagorean inequality work for triangle classification?

The Pythagorean theorem relates to right triangles. When $ c^2 , the largest angle is less than \( 90^\circ $ (acute). When $ c^2 > a^2 + b^2 $, it's greater than $ 90^\circ $ (obtuse).

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine whether the triangle is acute

00:04 Draw the perpendicular to our angle

00:14 We observe that our angle is larger than a right angle

00:31 Apply the same method to another angle

00:40 We observe that it's smaller than the right angle

00:47 A triangle with an angle greater than 90° is an obtuse triangle

00:51 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Is the triangle in the drawing an acute-angled triangle?

2

Step-by-step solution

To determine if the triangle is an acute-angled triangle, we need to understand the nature of its angles. In an acute-angled triangle, all three angles are less than $90^\circ$ . However, we do not have explicit angle measures or side lengths shown in the drawing. Instead, we assess the probable nature of the depicted triangle.

Given that an acute-angled triangle must have its largest angle smaller than $90^\circ$ , comparison property of triangle sides through Pythagorean type logic suggests that an acute triangle inequality $c^2 < a^2 + b^2$ (for sides $a$ , $b$ , and hypotenuse $c$ ) must hold.

In our problem, the depiction ultimately leads us to infer the implied relations among the triangle's angles. The given solution and analysis indicate it does not meet this criterion.

Hence, the triangle in the given drawing is not an acute-angled triangle, confirming the choice: No.

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic

Definition: Acute triangles have all three angles less than $90^\circ$
Method: Use inequality $c^2 < a^2 + b^2$ where c is longest side
Check: Visual inspection confirms at least one obtuse angle present ✓

Common Mistakes

Avoid these frequent errors

Assuming triangles look acute when drawn
Don't judge triangle types by visual appearance alone = wrong classification! Drawings can be misleading or not to scale. Always verify using angle measures or the Pythagorean inequality test for accurate classification.

Practice Quiz

Test your knowledge with interactive questions

In a right triangle, the side opposite the right angle is called....?

FAQ

Everything you need to know about this question

How can I tell if a triangle is acute without measuring angles?

+

Use the acute triangle test: If $c^2 < a^2 + b^2$ where c is the longest side, the triangle is acute. If $c^2 = a^2 + b^2$ , it's right. If $c^2 > a^2 + b^2$ , it's obtuse.

What makes a triangle NOT acute?

+

A triangle is not acute if it has even one angle that is $90^\circ$ or greater. Having just one right angle or obtuse angle disqualifies it from being acute.

Can I trust what triangles look like in drawings?

+

Never rely on visual appearance alone! Drawings may not be to scale or might be misleading. Always use mathematical tests or given measurements to classify triangles accurately.

What's the difference between acute, right, and obtuse triangles?

+

Acute: All angles < $90^\circ$
Right: One angle = $90^\circ$
Obtuse: One angle > $90^\circ$
Remember: A triangle can only be one type!

Why does the Pythagorean inequality work for triangle classification?

+

The Pythagorean theorem relates to right triangles. When $c^2 < a^2 + b^2$ , the largest angle is less than $90^\circ$ (acute). When $c^2 > a^2 + b^2$ , it's greater than $90^\circ$ (obtuse).

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