jugoslav

In the right triangle, ABC, angle B=90 degrees, and D and E lie on AC such that BD is a median and BE  is an altitude. If BD=3 x DE, compute AB/EC.

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The median of a triangle is a line drawn from one of the vertices to the mid-point of the opposite side. In the case of a right triangle, the median to the hypotenuse has the property that its length is equal to half the length of the hypotenuse.

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AD = CD = BD = 3            DE = 1/3(BD) = 1           EC = 2

BE = sqrt(3² - 1²) = √8

BC = sqrt(8 + 2²) = √12

AB = sqrt(AC² - BC²) = √24

AB / EC = √24 / 2

In triangle ABC, altitudes AD, BE, and CF intersects at the orthocenter H. If angle ABC = 49 and angle ACB = 21, then find the measure of BHC, in degrees. 

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Angle BHC = 49 + 21 = 70 degrees

A square sheet of paper ABCD is folded such that B falls on the midpoint M of CD. In what ratio does the crease divide BC?

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The ratio is 21 / 35

https://web2.0calc.com/questions/question_139  

(This excellent work has been done by heureka)

Angle   β = tan^-1(1/3) = 18.435º

Area = [10 * cos(β)]² = 90 squared units

In a triangle, ABC,  angle bisectors  AD, CF and BE meet at a point I. If DI=3, BD=4, and BI=5, then compute the area of triangle ABC.

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The following is what we know:

DI = 3           BD = 4          BI = 5

FI = EI = DI = 3

∠AEB ≅ ∠CEB = 90º            ∠IBD = tan^-1(3 / 4)  

This problem can be easily solved by using this information. 

Have fun.

https://web2.0calc.com/questions/in-triangle-abc-ab-5-ac-6-and-bc-7-circles-are-drawn

In right triangle ABC, AC = 8, AB = 15, and BC = 17.  Square ADEF is inscribed in the square.  Find the side length of square ADEF.

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sin(B) = 8 / 17 = 0.470588235

sin(C) = 15 / 17 = 0.882352941

(x / sinB) + (x / sinC) = 17     ==>  x = 5.217391

The perimeter of a triangle      P = 12√3 + 12

In triangle PQR, PQ = 13, QR = 14, and PR = 15. Let M be the midpoint of line QR. Find PM.

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a = 13        b = 14        c = 15

Median     PM = [sqrt(2a² + 2c² - b²)] / 2  ==>  PM = 12.166

It wasn't that hard.