SOLUTION: A town map shows the location of its school (S), municipal hall (H), church (C), and market (M). When these places are combined they form the shape of an isosceles trapezoid with a

Question 1205403: A town map shows the location of its school (S), municipal hall (H), church (C), and market (M). When these places are combined they form the shape of an isosceles trapezoid with an area of 475 sq. meters. The school (S) is adjacent to the municipal hall (H) and diagonally across to the church (C) (Note: SH is one of the non-parallel sides). The school (S) is 16 meters away from the market (M) and the municipal hall (H) is 22 meters away from the church (C). Suppose your house (E) is opposite the school and located in between the municipal hall and church, then how far is it from the school? Provide Illustration, Working Equation, Solution, and Answer.
Found 3 solutions by Theo, greenestamps, math_tutor2020:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
you state that the house (E) is opposite the school and located in between the municipal hall (H) and the church (C).

you don't state where E is between H and C.
i'll assume it's exactly in the middle.
that would be point E that divides HC into two equal parts that are HE and EC.
that tells you that HE = EC = 11.

i also drew a vertical line between SM and HC.
that would be point D that divides SM into two equal parts that are SD and DM.
that tells you that SD = DM = 8.

i also drew SE which the direct distance between point E (your house) and point S (the school) and i drew ME which is the direct distance between point E and point M (the market).

the vertical line is DE which is the altitude of the trapezoid.

the area of an isosceles trapezoid is equal to the sum of the two parallel sides divided by 2 * the height.

that would be (SM + HC) / 2 * DE
since the area is equal to 475 square meters and SM = 16 and HC = 22, you get:
475 = (16 + 22) / 2 + DE
solve for DE to get DE = 475 * 2 / (16 + 22) = 25.

the height of the trapezoid is DE whihc is equal to 25.

SE and SD form the right triangle SDE.
ME and MD form the right triangle MDE.
both SE and ME are equal to sqrt(8^2 + 25^2) = 26.2488095 meters.

the distance from point E to the school (point S) is equal to 26.2488095 meters.
that's your answer, assuming point E is equidistant from points H and C.

here's my diagram.

Answer by greenestamps(13209)   (Show Source): You can put this solution on YOUR website!

The problem is faulty, as tutor @Theo points out in his response. We know that E is between H and C; but we don't know WHERE between H and C it is.

So it is obviously impossible to know how far E is from S.

Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

I interpret the phrasing "Suppose your house (E) is opposite the school" to mean "points E and S are on the same vertical line".

S = school
H = municipal hall
C = church
M = market
E = your house

My drawing is almost the same as tutor Theo's drawing, except point E has been moved to the left to be directly under point S. Also, point D is not needed.

Recall the area of a trapezoid is:
area = 0.5*height*(base1+base2)
where base1 and base2 are the parallel sides.

In this case, SM = 16 and HC = 22 are the parallel sides.
The goal is to find the distance from your house (E) to your school (S).
In other words, we want to find the length of segment ES.
But notice that segment ES is the height of the trapezoid.
x = height = length of ES

area = 0.5*height*(base1+base2)
area = 0.5*ES*(SM+HC)
475 = 0.5*x*(16+22)
I'll let the student take over from here.

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