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Construct inscribed and circumscribed figures

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51219410
Construct a regular octagon inscribed in a circle. a) Find the octagon’s central angle. b) Describe how to locate all eight vertices by first drawing two perpendicular diameters and then constructing angle bisectors. c) Find the measure of one interior angle of the octagon.

Hints

- What angle is formed by two perpendicular lines? - What happens when each right angle at the center is bisected? - How are an interior angle and its adjacent exterior angle related?

Solution

1. The central angle is \(360^\circ \div 8 = 45^\circ\). 2. Draw two perpendicular diameters. Their endpoints give four points on the circle separated by \(90^\circ\) central angles. Bisect each of the four right angles at the center. The bisectors meet the circle at four additional points, producing eight equally spaced vertices separated by \(45^\circ\). 3. A regular octagon’s exterior angle equals its central angle, \(45^\circ\). Therefore, one interior angle is \(180^\circ - 45^\circ = 135^\circ\).

Answer

a) \(45^\circ\) b) Draw two perpendicular diameters, then bisect each \(90^\circ\) central angle to create eight equally spaced points. c) \(135^\circ\)
51280910
Use a circle with a diameter to construct right triangle \(ABC\) with hypotenuse \(AB = c = 7\,\text{cm}\) and leg \(BC = a = 4\,\text{cm}\).

Hints

- The right-angle vertex must lie on the circle whose diameter is the hypotenuse. - Use the given leg length to draw a second circle. - The intersections locate the possible positions of \(C\).

Solution

1. Draw \(\overline{AB}\) with \(AB = 7\,\text{cm}\). 2. Construct the midpoint \(M\) of \(\overline{AB}\). 3. Draw the circle centered at \(M\) with radius \(3.5\,\text{cm}\), so \(\overline{AB}\) is a diameter. 4. Draw a circle centered at \(B\) with radius \(4\,\text{cm}\). 5. Label either intersection of the two circles as \(C\), then draw \(\overline{AC}\) and \(\overline{BC}\). 6. The two intersections give reflected, congruent triangles. In either triangle, \(\angle C = 90^\circ\) because it subtends diameter \(\overline{AB}\).

Answer

Point \(C\) is either intersection of the circle with diameter \(\overline{AB}\) and the circle centered at \(B\) with radius \(4\,\text{cm}\). The two constructions are reflected, congruent solutions.
51271210
In isosceles triangle \(ABC\) with base \(\overline{AB}\), the vertex angle is \(\gamma=50^\circ\). The angle bisectors of the base angles \(\alpha\) and \(\beta\) intersect at \(W\). a) Find \(\alpha\) and \(\beta\). b) Find \(\angle AWB\). c) What special property does point \(W\) have in relation to the three sides of the triangle?

Hints

- Use the triangle angle sum. - In an isosceles triangle, the base angles are congruent. - An angle bisector divides an angle into two congruent angles. - What distance property does a point on an angle bisector have?

Solution

1. Since the triangle is isosceles, \(\alpha=\beta\). Using the triangle angle sum, \(\alpha=\beta=\frac{180^\circ-50^\circ}{2}=65^\circ\). 2. In triangle \(ABW\), the angles at \(A\) and \(B\) are half of the base angles, so each measures \(32.5^\circ\). 3. Therefore, \(\angle AWB=180^\circ-32.5^\circ-32.5^\circ=115^\circ\). 4. Since \(W\) is the intersection of the triangle's angle bisectors, it is the incenter. It is equidistant from all three sides.

Answer

a) \(\alpha=\beta=65^\circ\). b) \(\angle AWB=115^\circ\). c) \(W\) is the incenter and is the same perpendicular distance from all three sides.
51271310
A rectangular park \(ABCD\) has side lengths \(AB=12\,\text{m}\) and \(BC=5\,\text{m}\). Diagonal \(AC=13\,\text{m}\) divides the park into triangles \(ABC\) and \(ADC\). A fountain will be placed at the incenter of each triangle. a) Explain how angle bisectors determine the location \(W_1\) of the fountain in triangle \(ABC\). b) For a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the inradius is \(r=\frac{a+b-c}{2}\). Find the distance from \(W_1\) to side \(\overline{AB}\). c) How far is each fountain, \(W_1\) and \(W_2\), from diagonal \(\overline{AC}\)? Justify your answer.

Hints

- How do you locate the center of a circle tangent to all three sides of a triangle? - What does the inradius measure? - Compare the two triangles formed by the diagonal of the rectangle.

Solution

1. Construct at least two angle bisectors of triangle \(ABC\). Their intersection is the incenter \(W_1\). 2. The perpendicular distance from an incenter to any side equals the inradius. Here, \(r=\frac{12+5-13}{2}=\frac{4}{2}=2\,\text{m}\). 3. Triangles \(ABC\) and \(ADC\) are congruent \(5\)-\(12\)-\(13\) right triangles, so their incircles have the same radius. 4. Since \(\overline{AC}\) is a side of each triangle, each incenter is \(2\,\text{m}\) from \(\overline{AC}\).

Answer

a) \(W_1\) is the intersection of the angle bisectors of triangle \(ABC\). b) \(2\,\text{m}\). c) Each fountain is \(2\,\text{m}\) from \(\overline{AC}\).

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