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Construct the incircle of the triangle ABC below using only a compass and straight edge - Leaving Cert Mathematics - Question 3 - 2010

Question 3

Construct the incircle of the triangle ABC below using only a compass and straight edge. Show all construction lines clearly. An equilateral triangle has sides of l... show full transcript

Worked Solution & Example Answer:Construct the incircle of the triangle ABC below using only a compass and straight edge - Leaving Cert Mathematics - Question 3 - 2010

Step 1

Construct the incircle of the triangle ABC using only a compass and straight edge.

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Answer

Draw Triangle ABC: Start by drawing the triangle ABC on a piece of paper using a straight edge.
Construct Angle Bisectors: Use a compass to find the midpoint of segments AB and AC. Draw arcs centered at points A and B of equal radii that intersect at two points. Connect these intersection points to form the angle bisector of angle A. Repeat this process for angle B.
Locate the Incenter: The intersection point of the angle bisectors is the incenter of triangle ABC, which will be labeled as point I.
Draw the Perpendicular from Incenter: From point I, draw a perpendicular line to side BC. The intersection point on BC will be labeled as D.
Construct the Incircle: With a compass set to the distance from I to D (the radius), draw a circle centered at point I. This circle is the incircle, touching all three sides of the triangle ABC.
Mark and Label Points Clearly: Clearly mark all points, including the incenter I and points A, B, C, and D on the diagram for clarity.

Step 2

Find the area of its incircle.

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Answer

To find the area of the incircle of an equilateral triangle with sides of length 2 units, we first need to find the radius of the incircle (r). The formula to find the radius of the incircle for an equilateral triangle is:

$r = \frac{a}{2\sqrt{3}}$

where 'a' is the length of a side. Substituting in:

$r = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}}$

Next, the area (A) of the incircle can be found using the formula:

$A = \pi r^2$

Substituting for r:

$A = \pi \left(\frac{1}{\sqrt{3}}\right)^2 = \pi \left(\frac{1}{3}\right) = \frac{\pi}{3}$

Thus, the area of the incircle is ( \frac{\pi}{3} ) square units.

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Construct the incircle of the triangle ABC below using only a compass and straight edge - Leaving Cert Mathematics - Question 3 - 2010

Worked Solution & Example Answer:Construct the incircle of the triangle ABC below using only a compass and straight edge - Leaving Cert Mathematics - Question 3 - 2010

Construct the incircle of the triangle ABC using only a compass and straight edge.

Find the area of its incircle.

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