Question

a walk in the park
a civil engineering firm is designing a formal garden in the town
park. the firm has drawn scaled plans for some of the paths and
hedges in the circular garden. paths ec and db are both
diameters. use the information to help develop the rest of the
blueprint.
question
part a
using the designers current blueprint, identify these angle
and arc relationships in the park.
select the correct answer from each drop - down menu.
∠eda is intercepting ea.
∠eyb is intercepting eb.
question
part b
if someone walks along the outside of the garden from
point a to point b, what percent of the gardens border
would they have walked around? round your answer to the
nearest whole percent.
type the correct answer in the box. use numerals instead of
words.
they would have walked around approximately
% of the outside border of the garden.
part c
in the designers proposal, they state that paths ec and db
each measure 100 feet. they also state that the garden will
require 140 feet of hedges. are their numbers reasonable?
explain your reasoning.

Explanation:

Part A

For \(\angle EDA\)

• Step1: Recall angle - arc relationship

An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle. The measure of an inscribed angle is half of the measure of the intercepted arc. \(\angle EDA\) has its vertex on the circle and intercepts arc \(EA\). So, \(\angle EDA\) is an inscribed angle.

For \(\angle EYB\)

• Step1: Recall angle - arc relationship

A central angle is an angle whose vertex is at the center of the circle. \(\angle EYB\) has its vertex at the center \(Y\) of the circle (since \(EC\) and \(DB\) are diameters and \(Y\) is the intersection point of the diameters, and in a circle, the intersection of two diameters is the center) and intercepts arc \(EB\). So, \(\angle EYB\) is a central angle.

Part B

• Step1: Find the measure of arc \(AB\)

The sum of angles around a point is \(360^{\circ}\). We know that \(\angle AXE = 70^{\circ}\) (given). The measure of arc \(AE\) is \(50^{\circ}\) (given). Let the measure of arc \(AB\) be \(x\).
Since \(\angle AXE\) is a vertical angle to the angle opposite to it formed by the intersection of the chords. Using the property that the sum of arcs in a circle is \(360^{\circ}\), and if we assume the circle has a total of \(360^{\circ}\) of arc length.
We know that the measure of an arc is related to the central - angle that intercepts it.
The measure of arc \(AB\):

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• Step2: Calculate the percentage

The formula for the percentage of an arc length (or the portion of the circumference) is \(\text{Percentage}=\frac{\text{Measure of arc}}{\text{Total measure of circle}(360^{\circ})}\times100\)

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Part C

• Step1: Recall the property of a circle

In a circle, the diameter is the longest chord. If \(EC\) and \(DB\) are diameters (given in the problem statement), then the length of any chord (in this case, the hedges) must be less than or equal to the length of the diameter.
If \(EC = DB=100\) feet (diameters), then the length of any chord (hedges) should be \(\leq100\) feet. But the designer claims that the garden will require \(140\) feet of hedges. Since \(140>100\), their numbers are not reasonable.

Answer:

Part A

• \(\angle EDA\) is an inscribed angle intercepting \(\overset{\frown}{EA}\).
• \(\angle EYB\) is a central angle intercepting \(\overset{\frown}{EB}\).

Part B

They would have walked around approximately \(17\%\) of the outside border of the garden.

Part C

No, their numbers are not reasonable. Because in a circle, the diameter is the longest chord. If \(EC\) and \(DB\) (diameters) are \(100\) feet, the length of hedges (a chord) cannot be \(140\) feet since \(140>100\).