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Elan Coil [88]
1 year ago
7

Triangle PQR is formed by connecting the midpoints of the side of triangle MNO. The lengths of the sides of triangle PQR are sho

wn. What is the length of MN? Figures not necessarily drawn to scale.

Mathematics
1 answer:
stiv31 [10]1 year ago
7 0

The length of segment \overline{MN} found using properties of an equilateral triangle and segment addition postulate is 10

<h3>What is segment addition postulate?</h3>

Segment addition postulate states that a line AC contains a point B if we have AB + BC = AC

The sides of ∆PQR are equal to 5, therefore, ∆PQR is an equilateral triangle

According to the mid segment theorem, we have;

RQ || MN

RQ = 0.5 × MN

RP || ON

RP = 0.5 × ON

PQ || MO

PQ = 0.5 × MO

Therefore, from alternate interior angles theorem, we have;

‹PRQ = ‹RPM

‹PQR = ‹QPN

‹QRP = ‹RQO

However, ‹PRQ = ‹PQR = ‹QPR = 60°, which gives;

∆MRP and ∆QPN are equilateral triangles of side length 5

From which we have;

MR = RP = MP = 5

PN = QN = PQ = 5

MN = MP + PN; Segment addition postulate

Therefore;

MN = 5 + 5 = 10

Learn more about segment addition postulate here:

brainly.com/question/1721582

#SPJ1

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Step-by-step explanation:

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Direction of v: α = tan⁻¹|6/-1| = tan⁻¹|-6| = tan⁻¹(6) = 80.53° which is your reference angle, but to verify that the angle is in the second quadrant, you'll need to do θv = 180° - 80.53° = 99.47°, therefore your direction angle is θv=100° to the nearest degree.

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3 years ago
This week, we are covering relationships that can be approximated by linear equations. For instance, y = 453x + 3768 represents
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Answer:

See explanation below.

Step-by-step explanation:

We assume that the data is given by :

x: 30, 30, 30, 50, 50, 50, 70,70, 70,90,90,90

y: 38, 43, 29, 32, 26, 33, 19, 27, 23, 14, 19, 21.

Where X represent the cost for scholarships in thousands of dollars and y represent the cost of life for an academic semester (The data comes from the web)

We can find the least-squares line appropriate for this data.  

For this case we need to calculate the slope with the following formula:

m=\frac{S_{xy}}{S_{xx}}

Where:

S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}

S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}

So we can find the sums like this:

\sum_{i=1}^n x_i = 30+30+30+50+50+50+70+70+70+90+90+90=720

\sum_{i=1}^n y_i =38+43+29+32+26+33+19+27+23+14+19+21=324

\sum_{i=1}^n x^2_i =30^2+30^2+30^2+50^2+50^2+50^2+70^2+70^2+70^2+90^2+90^2+90^2=49200

\sum_{i=1}^n y^2_i =38^2+43^2+29^2+32^2+26^2+33^2+19^2+27^2+23^2+14^2+19^2+21^2=9540

\sum_{i=1}^n x_i y_i =30*38+30*43+30*29+50*32+50*26+50*33+70*19+70*27+70*23+90*14+90*19+90*21=17540

With these we can find the sums:

S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=49200-\frac{720^2}{12}=6000

S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}=17540-\frac{720*324}{12}{12}=-1900

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m=-\frac{1900}{6000}=-0.317

Nowe we can find the means for x and y like this:

\bar x= \frac{\sum x_i}{n}=\frac{720}{12}=60

\bar y= \frac{\sum y_i}{n}=\frac{324}{12}=27

And we can find the intercept using this:

b=\bar y -m \bar x=27-(-0.317*60)=46.02

So the line would be given by:

y=-0.317 x +46.02

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